LECTURES ON QUOTIENTS AND MODULI SPACES
YI HU
Department of Mathematics
University of Arizona
yhu@math.arizona.edu
1
C ONTENTS
0. Introduction
5
1. Ideas of Quotient Theories
5
1.1. What are Moduli Spaces?
5
1.2. What are GIT quotients?
6
1.3. What are Chow Quotients
8
2. Basics of Geometric Invariant Theory
10
2.1. Algebraic groups
10
2.2. Group actions and basic results
12
2.3. Categorical, geometric and good quotients
15
2.4. The Hilbert-Nagata theorem
19
2.5. Quotients of affine schemes
22
2.6. Linearization and stability
25
2.7. Existence of GIT quotients
30
3. Criteria for Stability
35
3.1. A geometric criterion
35
3.2. The Hilbert-Mumford numerical criterion
38
3.3. The Kempf-Ness analytic criterion
45
3.4. Moment map
49
3.5. The moment map criterion
52
3.6. Relation with symplectic reduction
53
4. Configuration of linear subspaces
55
4.1. Criteria for stable configuration
55
4.2. Harder-Narasimhan and Jordan-Hölder filtrations
61
4.3. Polystable configurations
65
4.4. Balanced metrics and polystable configurations
67
4.5. Stability of tensor product
69
4.6. Generalized Gelfand-MacPherson correspondence
70
4.7. The Cone of effective linearizations
73
5. Stratification of the Set of Unstable Points
77
L
5.1. The function M (x) and the numerical criterion revisited
77
5.2.
Adapted one-parameter subgroups
81
5.3.
Stratification of the set of unstable points
83
5.4. Finiteness theorems
85
6. Moment Maps and GIT
87
6.1. Quantization commutes with reduction
87
6.2. GIT quotients of X × G/B and general symplectic reductions.
88
6.3. Rational points in the moment map image
90
L
6.4. The function M (x) and moment map
90
6.5. Homological equivalence for G-linearized line bundles
91
2
7.
G-ample cone: Walls and chambers
97
7.1.
G-effective line bundles
97
7.2.
The G-ample cone
99
7.3.
Walls and chambers
103
7.4. GIT-equivalence classes
111
7.5.
Abundant actions
113
8. Variation of GIT quotients
114
8.1. Faithful walls
114
8.2. GIT wall-crossing flips
115
9. Geometric Invariant Theory and Birational Geometry
122
9.1. GIT realization of Birational morphism
122
9.2. Partial desingularization of singular quotient
123
9.3. Weak Factorization Theorem
123
9.4. Proof of Theorem 9.3.1
125
9.5. GIT on Projective Varieties with Finite Quotient Singularites
126
10. Chow Quotient
127
10.1. Chow Variety
127
10.2. Definition of Chow quotient
129
10.3. The Chow family and Chow cycles
130
10.4. Chow quotient dominates GIT quotients
132
10.5. Relation with the limit quotient
133
10.6. Ample line bundles over the Chow quotient
135
10.7. Perturbing, translating and specializing
137
10.8.
140
M 0,n
11. Moduli Spaces of Coherent Sheaves
141
11.1. Stabilities of coherent sheaves
141
11.2. Quot scheme and Grothendieck embedding
141
11.3. Moduli of semistable coherent sheaves
142
12. Balanced Configuration and Moment Map
143
12.1. Quot scheme and Hom(X, Gr)
143
12.2. Moment map for singular varieties
144
N
145
12.3. Moment map for Hom(X, Gr(r, C ); P )
12.4. Balanced Configuration and Stability
146
13. Relative GIT and Universal Moduli
147
14. Stabilities of Polarized Projective Variety
148
14.1. Stabilities on Hilbert and Chow points
148
14.2. Moment map and balanced metric
149
14.3. Yau’s program on stability and special metrics
149
15. Moduli Stacks
150
15.1. Overview on stacks
154
16. Quotients as algebraic spaces
156
3
16.1. Proper actions
156
16.2. Existence of quotients
156
16.3. Kollár’s projective criterion
156
16.4. Universal moduli of sheaves over M g,n
156
17. Moduli Functors
156
18. Representable Moduli Functors
158
19. Properties of Moduli Functors
159
20. Moduli Space of Stable Maps
160
21. Mirror Principles
160
References
161
4
0. I NTRODUCTION
Most of the time we will only provide proofs over C, even though much
of the results are also valid over more general base fields.
1. I DEAS
OF
Q UOTIENT T HEORIES
1.1. What are Moduli Spaces?
A moduli space is, roughly speaking, a parameter space for equivalence
classes of certain geometric objects of fixed topological type. Depending
on purposes, there are two types of moduli: fine moduli and coarse moduli. The former often does not exist as a (quasi-) projective scheme but
lives naturally as a stack, while the latter frequently exists as a (quasi-)
projective scheme.
A naive way to think of a moduli space M is
M = {Geometric Objects}/ ∼
as a collection of the geometric objects of the interest modulo equivalence
relation. This way of thinking is “coarse”. Such a moduli space M often
exists as a projective scheme.
To describe fine moduli, one needs moduli functor, a categorical language for the problem. A moduli functor is covariant functor
F : {schemes} → {sets}
which sends any scheme X to a family of the geometric objects parameterized by X. The moduli space M is fine, or the moduli functor F is
represented by M if there is a universal family over M
U →M
such that for any scheme X and a family of the geometric objects
Z→X
parameterized by X, it corresponds to a morphism f : X → M so that the
family Z → X is the pullback of the universal family via the morphism f
Z = f ∗ U −−−→


y
X
U


y
f
−−−→ M
Fine moduli in general does not exist if some objects possess nontrivial automorphisms. Therefore we are somehow forced to consider coarse
moduli if we prefer to work on projective schemes.
5
Now let us return to a coarse moduli as a parameter space of geometric
objects
M = {Geometric Objects}/ ∼ .
This way of thinking, as it stands, naturally lead to a quotient theory. First
we denote the collection of the objects as
X = {Geometric Objects}
and then we would introduce a natural group action
G×X →X
such that two geometric objects x, y are equivalent if and only if they, as
points in X, are in the same group orbit:
x ∼ y if and only if G · x = G · y.
Then the moduli space can be taken as the quotient space of G-orbits
“X/G.”
This is what the general idea is. However, taking quotients in algebraic
geometry can be very subtle. Indeed, there are several theories about this:
•
•
•
•
The Hilbert-Mumford Geometric Invariant Theory;
Chow quotient varieties
Hilbert quotient varieties;
Artin, Kollár, Keel-Mori quotient spaces.
We will only focus on the first two: GIT quotients and Chow quotients.
1.2. What are GIT quotients?
To give the reader some intuitive ideas about GIT quotients, let us informally consider a simple, yet quite informative “toy” example.
Let G = C∗ act on P2 by
λ · [x : y : z] = [λx : λ−1 y : z].
Consider a map Φ : P2 → R given by
Φ([x : y : z]) =
|x|2 − |y|2
.
|x|2 + |y|2 + |z|2
This is the so-called moment map for the induced symplectic S 1 -action
with respect to the Fubibi-Study metric. Its image is the interval [−1, 1].
6
[0:0:1]
Y=0
X=0
[0:1:0]
[1:0:0]
Z=0
XY = a Z 2
Φ
−1
0
1
Figure 1. Conics
∗
The C orbits are classified as follows. (See Figure 1 for an illustration.)
• Generic C∗ -orbits are conics XY = aZ 2 minus two points [1:0:0]
and [0:1:0] for a 6= 0, ∞. We denote these orbits by (XY = aZ 2 ).
The moment map image of the orbit (XY = aZ 2 ) is (−1, 1).
• Other 1-dimensional orbits are the three coordinate lines X = 0,
Y = 0, and Z = 0 minus the coordinate points on them. We denote
these orbits by (X = 0), (Y = 0), and (Z = 0).
The moment map images of the orbits (X = 0), (Y = 0), and
(Z = 0) are (−1, 0), (0, 1), and (−1, 1), respectively.
• Finally, the fixed points are the three coordinate points, [1:0:0], [0:1:0],
and [0:0:1].
Their moment map images are 1, −1, and 0, respectively.
The reader can easily check that the ordinary topological orbit space
P2 /C∗ is a nasty non-Hausdorff space. For example, the orbits (X = 0) and
(Y = 0) are both contained in the limit of the generic orbits (XY = aZ 2 )
when a approaches 0, thus can not be separated.
The idea of GIT is to find some open subset U ⊂ P2 , called the set of
semistable points, such that a good quotient in a suitable sense, U//C∗ ,
exists. In this particular example, there are three open subsets that admit
good quotients.
U[−1,0] = P2 \ {Y = 0},
U[0,1] = P2 \ {X = 0},
7
U{0} = P2 \ {[1 : 0 : 0], [0 : 1 : 0]}.
The meaning of the indexes of these three open subsets are as follows.
The moment map Φ has three critical values −1, 0, and 1 which divide the
interval into two top chambers [−1, 0] and [0, 1], and three 0-dimensional
chambers {−1}, {0}, {1}. Each chamber C defines a GIT stability: a point
[x : y : z] is semi-stable with respect to the chamber C if and only if
C ⊂ Φ(C∗ · [x : y : z]),
and it is stable if the (relative) interior C ◦ of C satisfies
C ◦ ⊂ Φ(C∗ · [x : y : z]) and dim C∗ · [x : y : z] = 1.
(For a reference for this, see for example, [33].) One checks readily that
each of the above three open subsets are semistable locus with respect to
the corresponding chamber. Thus, for example, the orbit (X = 0) is stable
with respect to [−1, 0], unstable with respect to [0, 1]; while the orbit (Y =
0) is stable with respect to [1, 0], unstable with respect to [−1, 0]. But, (X =
0), (Y = 0), and [0 : 0 : 1] are all semi-stable with respect to the chamber
{0}. Finally, observe that the generic orbits, namely the conics (XY = aZ 2 )
are stable with respect to any chamber.
A GIT quotient in general does not parameterize orbits. But, it should
always parameterize orbits that are closed in the semi-stable locus. Here
in this example, the GIT quotient X[−1,0] defined by the chamber [−1, 0]
parameterizes (XY = aZ 2 ) (a 6= 0, ∞), (Z = 0), and (X = 0). Thus
X[−1,0] is isomorphic to P1 with (Z = 0) and (X = 0) serve as ∞ and 0,
respectively. Likewise, the GIT quotient X[0,1] defined by the chamber [0, 1]
parameterizes (XY = aZ 2 ), (Z = 0), and (Y = 0). And, the GIT quotient
X{0} defined by the chamber {0} parameterizes (XY = aZ 2 ), (Z = 0), and
[0 : 0 : 1].
All these quotients are isomorphic to P1 , and hence they are all isomorphic to each other. This is very special because the dimension is too low
(namely 1) to allow any variation. In general, they should be quite different and are only birational to each other (see §2.1). This was a very decisive observation, and the determination to investigate the most general
relation among different GIT quotients led to some significant discoveries
and applications.
1.3. What are Chow Quotients.
In the “toy” example, observe that the lines X = 0 and Z = 0, which are
of degree 1, have different homology classes than the conic orbits XY =
aZ 2 , which are of degree 2. But the GIT quotient X[−1,0] parameterizes
these orbits of different homology classes. Even worst, the two orbits (X =
0) and (Y = 0) are all identified with the closed orbit [0 : 0 : 1] in the
quotient X{0} , and the orbit [0 : 0 : 1] even has smaller dimension than the
8
dimension of the generic conic orbits. These are not desirable or suitable
for moduli problems.
This kind of problem can however be overcome by considering Chow
quotient which takes completely different approach.
Return to our “toy example”, to obtain the Chow quotient, we first consider the closures of the generic C∗ -orbits, XY = aZ 2 (a 6= 0, ∞), and then
looks at all their possible degenerations. When a = 0, we get the degenerated conic XY = 0, two crossing lines; and when a = ∞, we obtain Z 2 = 0,
a double line. They all have the same homology classes (degree 2). And
the Chow quotient is the space of all C∗ -invariant conics. (See Figure 1.) Each
point of the Chow quotient represents an invariant algebraic cycle. In this
case, the generic cycles are [XY = aZ 2 ] (a 6= 0, ∞), and the special cycles
are [X = 0] + [Y = 0] (a = 0), and 2[Z = 0] (a = ∞).
From the above example, we see that Chow quotient parameterizes cycles of generic orbit closures and their toric degenerations which are certain sums of orbit closures of dimension dim G. We will call these cycles Chow
cycles or Chow fibers.
So, when do two arbitrary points belong to the same Chow cycle?
Consider the example again. We have that [0 : y : z] and [x : 0 : z]
(xyz 6= 0) belong to the same Chow cycle XY = 0. To get [x : 0 : z] from
[0 : y : z], we first perturb [0 : y : z] to a general position
ϕ(t) = [tx : y : z](t 6= 0),
then translate it by g(t) = t−1 ∈ C∗ to
g(t) · ϕ(t) = [x : ty : z],
and then g(t) · ϕ(t) specializes to [x : 0 : z] when t goes to 0. We will call this
process perturbation-translation-specialization (PTS). It turns out this simple
relation holds true in general. That is, we prove in general that two points
x and y of X, with
dim G · x = dim G · y = dim G,
belong to the same Chow cycle if and only if x can be perturbed (to general positions), translated along G-orbits (to positions close to y), and then
specialized to the point y.
The general case will be treated in later sections.
9
2. B ASICS
OF
G EOMETRIC I NVARIANT T HEORY
One of ideas of constructing a general GIT quotient is to go from local
to global. That is, given a reductive algebraic group action on a projective variety X, take an equivariant embedding of X into some projective
space, or almost equivalently, take an ample line bundle L over X and lift
the action to the line bundle. Then use non-vanishing loci of equivariant
sections, we can cover an open subset of X by G-invariant affine open varieties. As it turns out, any G- affine variety always admits “good” affine
quotient. Therefore, the task left is to show that these affine quotients can
be patched together to form a global “good” quotient. In this book, we
will take this approach.
To begin with, we will first introduce algebraic groups, then their actions, and then quotients of affine varieties, followed by general global
GIT quotients.
Some important criteria for stability will be introduced and studied in
the next chapter.
2.1. Algebraic groups.
Let k be a fixed base field.
An algebraic group G is a group together with a structure of algberaic
scheme on G such that the group laws
G × G → G, (g, g 0 ) → gg 0 ,
G → G, g → g −1
are morphisms.
The standard example of an algebraic group is GL(n, k) or more generally the group GL(V ) of invertible linear transformations of a vector space
V . Given any algebraic morphism
ρ : G → GL(V )
which is also a group homomorphism, we obtain an action of G on V . Such
a homomorphism is called a (rational) representation of G.
An algebraic group which is isomorphic to a closed subgroup of GL(n)
is called a linear algebraic group. It is automatically affine. Conversely, any
affine algebraic group is linear ([8]).
Any linear algebraic group has a unique maximal connected normal
solvable subgroup, called its radical. G is reductive if its radical is a (split)
torus, i.e., isomorphic to a direct product of k ∗ = k \ {0}.
A linear algebraic group is called geometrically reductive (resp. linearly
reductive) if, for every linear action of G on k n , and every invariant point
10
v of k n , v 6= 0, there is an invariant homogeneous polynomial f of degree
≥ 1 (resp. = 1) such that f (v) 6= 0.
G is linearly reductive if and only if every (rational) representation is
completely reducible, i.e., it is a direct sum of some irreducible representations.
Over characteristic 0, all three concepts are equivalent. Over an arbitrary field, reductive is equivalent to geometrically reductive, thanks to
Haboush’s solution of Mumford’s conjecture. Over positive characteristics, Nagata showed that the only connected groups that are linearly reductive are (split) tori.
Exercises
1. Prove that any algebraic group is smooth.
2. Prove that there is no non-trivial homomorphism from Gm,k to Ga,k or
from Ga,k to Gm,k . Here Gm,k is the multiplicative group k ∗ and Ga,k is the
additive group k.
3. Show that Ga,k is not geometrically reductive. (Hint: Consider the action
on V = A2k by the formula t · (x, y) = (x, y + tx).)
11
2.2. Group actions and basic results.
An action of an algebraic group G on a scheme is a morphism
σ :G×X →X
such that if writing σ(g, x) = g · x, then
g(g 0 · x) = (gg 0 ) · x, and e · x = x
where e is the identity element of G. For a given point x ∈ X, the stabilizer
Gx of x is the closed subgroup
Gx = {g ∈ G : g · x = x}.
Alternatively, it is the fiber over x of the morphism
ϕx : G → G · x ,→ X
g → g · x.
As explained in §1, there are in general many open subsets of X that
can have “good” quotients. For each such open subset, to construct the
quotient space as an algebraic variety, the route to the goal is from local
to global. That is, we will first construct affine quotients, then patch them
together to form a global quotient. Hence, we begin with actions on affine
varieties.
For a scheme X, let O(X) be the algebraic of regular functions. For affine
variety, this is isomorphic to the coordinate algebra. When both G and X
are affine, an action
σ :G×X →X
is equivalent to a coaction homomorphism
σ ∗ : O(X) → O(G) ⊗ O(X).
This defines a unique action of G on O(X). One can check that if we denote
g · h by hg for any g ∈ G and h ∈ O(X), then hg (x) = h(g · x) for all x ∈ X.
Lemma 2.2.1. Assume that G is an affine algebraic group acting on a scheme X.
Let W be a finite-dimensional subspace of O(X). Then we have
(1) if W is G-invariant, then the action of G on W is given by a (rational)
representation;
(2) W is always contained in a finite-dimensional invariant subspace of O(X).
Proof. (1). Let h1 , . . . , hn be a basis of W . For any function h ∈ O(X) and
g ∈ G, recall that hg ∈ O(X) is given by hg (x) = h(g · x) for all x ∈ X. Then
we can write
X
hgi =
aij (g)hj ,
One can check that the formula
g → (aij (g))
12
defines a homomorphism
ρ : G → GL(n)
which is a rational representation.
(2). It suffices to prove that
W 0 = span{hgi : 1 ≤ i ≤ n, g ∈ G}
is of finite dimension. For this, define Hi ∈ O(G × X) by
Hi (g, x) = hi (g · x).
Since O(G × X) = O(G) ⊗ O(X), we can write
X
Hi =
Gij ⊗ Fij
with Gij ∈ O(G) and Fij ∈ O(X). Let W 00 be the subspace spanned by Fij .
Since
X
hgi (x) = Hi (g, x) =
Gij (g)Fij (x),
it follows that W 0 ⊂ W 00 . But W 00 is finite-dimensional, hence so is W 0 .
¤
Corollary 2.2.2. Let G be a geometrically reductive group acting on an affine
scheme X and W0 , W1 two disjoint invariant closed subsets of X. Then there is a
function f ∈ O(X)G such that f (W0 ) = 0 and f (W1 ) = 1.
Proof. First we choose h ∈ O(X) such that h(W0 ) = 0 and h(W1 ) = 1. Now
by Lemma 2.2.1,
W = span{hg : g ∈ G}
is of finite dimension. Let h1 , . . . , hn be a basis of this subspace, we can
write
X
hgi =
aij (g)hj ,
where
g → (aij (g))
defines a rational representation of G. This representation determines a
linear action of G on k n , which makes the morphism
ψ : X → k n , , ψ(x) = (h1 (x), . . . , hn (x))
into a G-morphism. Moreover, ψ(W0 ) = 0 and ψ(W1 ) = {v} with v 6= 0.
Because G is (geometrically) reductive, there exists a G-invariant homogeneous polynomial χ such that χ(v) = 1. Then f = χ ◦ h is an invariant
function in O(X) with the desired properties.
¤
Exercises
1. An action is closed if all orbits are closed. Show that an action is closed
if all the stabilizers are of the same dimension. What about the converse?
13
2. An action is proper if
Ψ:G×X →X ×X
(g, x) → (g · x, x)
is a proper morphism. Show that a proper action is closed.
3. Let Vn stand for n + 1-dimensional affine space of homogeneous forms
in x and y of degree n over C. Then G = SL(2) acts on Vn by means of
substitutions in x and y. Define X ⊂ V1 × V4 by (F1 , F4 ) ∈ X if (a) F1 6= 0;
(b) F4 is the square of a homogeneous quadratic form of discriminant 1.
Prove that
(1) X is non-singular and is invariant under SL(2).
(2) the isotropy subgroup of SL(2) at any point of X is trivial.
(3) the morphism Ψ is not closed, in particular, the action is not proper.
(Hint. Let Z be the closed subscheme of G × X whose points are the
pairs
¶
µ
0 −λ−1
× (λx + y, x2 y 2 )
λ
0
for λ ∈ C∗ . The image under Ψ is
(λx + y, x2 y 2 ) × (λx + y, x2 y 2 ).
In the closure of this set is the extra point
(y, x2 y 2 ) × (y, x2 y 2 ).)
4. Show that all stabilizer subgroups of a proper action of an affine algebraic group are finite. Show that the converse is not true.
5. Give an example of a projective G-variety of dimension greater than 1
which contains an open orbit with the complement a finite set.
6. Construct a counterexample to Corollary 2.2.2 when G is the additive
group Ga,k .
7. Let H be a closed subgroup of an algebraic group G. Use Lemma 2.2.1 to
show that G/H is quasi-projective. (Hint: Let I be the ideal of H in O(G)
and V the subspace of O(G) spanned by the G-translates of generators of
I. Let W = V ∩ I with n = dim W . Then G acts on the Grassmannian
variety Gr(n, V ).)
14
2.3. Categorical, geometric and good quotients.
Definition 2.3.1. Let G be an algebraic group acting on a scheme X. A
categorical quotient of X by G is a pair (Y, φ) where Y is a scheme with
the trivial G-action and
φ:X→Y
is a G-morphism such that for any scheme Z with the trivial G-action and
a G-morphism ψ : X → Z, there is a unique morphism χ : Y → Z such
that ψ = χ ◦ φ, that is, we have a commutative diagram
ψ
X −−−→ Z




φy
idy
χ
Y −−−→ Z.
Frequently, we write Y = X//G.
Definition 2.3.2. A categorical quotient is called a geometric quotient1 if
the image of Ψ : G × X → X × X is X ×Y X.
It follows from the definition that a categorical (hence also geometric)
quotient is unique up to canonical isomorphism.
As one can check easily, that the image of Ψ : G × X → X × X is X ×Y X
means that any fiber of φ consists of a single orbit and hence a geometric
quotient is the ordinary orbit space X/G (at least set-theoretically). Unlike
geometric quotient, a categorical quotient is in general not an ordinary
orbit space. In the “toy” example of §1, the quotients defined by the chambers [−1, 0] and [0, 1] are geometric, but the quotient defined by {0} is not.
We will see this formally from Example 2.5.2 later.
The notion of categorical quotient is the weakest one we will use. Quotients that GIT method provides actually satisfy stronger properties. As
we will show in the next section, for example, quotients of affine varieties
will enjoy the following list of properties which we now use as the definition of the so-called good quotient.
Definition 2.3.3. Let φ : X → Y be a surjective G-morphism where G acts
on Y trivially. (Y, φ) is called a good quotient of X by G if it satisfies
φ∗
(1) If U is open in Y , then O(U ) → O(φ−1 (U ))G is an isomorphism;
(2) If W is a closed invariant subset of X, then φ(W ) is closed in Y ;
(3) if W1 and W2 are disjoint invariant closed subsets of X, then φ(W1 )∩
φ(W2 ) = ∅.
1
GIT treats group actions exclusively in the category of schemes. In this case, a geometric quotient (however reasonably defined) is necessarily categorical. Our definition is
a short-cut as we will see later that a geometric quotient can fail to be categorical in the
category of algebraic spaces.
15
A good quotient is called a good geometric quotient if in addition the
image of Ψ : G × X → X × X is X ×Y X.
Proposition 2.3.4. A good quotient is necessarily a categorical quotient.
Proof. Let ψ : X → Z be a G-morphism where Z is a scheme with the
trivial G-action. We need to show that there is a morphism χ : Y → Z
such that χ ◦ φ = ψ:
ψ
X −−−→ Z




φy
idy
χ
Y −−−→ Z.
First ψ(φ−1 (y)) consists of a single point. Otherwise there are z1 , z2 ∈
ψ(φ−1 (y)) with z1 6= z2 . Let Wi = ψ −1 (zi ) (i = 1, 2), we obtain contradiction
to (3). Hence χ exists as a set-theoretic map.
If V is open in Z, by the surjectivity of φ, we can check that
χ−1 (V ) = Y \ φ(X \ ψ −1 (V ))
is open.
It remains to show that χ is a morphism. Note first that we have
image(ψ ∗ : O(V ) → O(ψ −1 (V )) ⊂ O(ψ −1 (V ))G
because ψ is a G-morphism. By (1), we have
∼
=
φ∗ : O(χ−1 (V )) −−−→ O(ψ −1 (V ))G .
Thus we have a homomorphism
(φ∗ )−1 ◦ ψ ∗ : O(V ) → O(χ−1 (V ))
and hence a morphism
χ0 : χ−1 (V ) → V.
Clearly, χ0 also satisfies
χ0 ◦ (φ|ψ−1 (V ) ) = ψ|ψ−1 (V ) .
Since φ is surjective, we obtain that
χ0 = χ|ψ−1 (V ) .
This completes the proof.
¤
Remark 2.3.5. As we see from the proofs,
• (3) is a property that guarantees χ exists as a set-theoretic map;
• (2) grants that χ is continuous;
• (1) makes it algebraic.
16
Proposition 2.3.6. Let X be a G-scheme. A good quotient
φ:X→Y
enjoys the following properties:
(1) two points x and x0 of X have the same image in Y if and only if G · x ∩
G · x0 6= ∅;
(2) for each point y ∈ Y , the fiber φ−1 (y) contains exactly one closed orbit.
Proof. (1). The direction “=⇒” follows directly from the condition (3) of
Definiton 2.3.3. Conversely, if x and x0 of X have different images in Y ,
then they lie in different fibers of φ. Since fibers are closed subsets of X,
G · x and G · x0 lie in different fibers. By Definiton 2.3.3 (3) again, G · x ∩
G · x0 = ∅;
(2). By (1), two closed orbits in the fiber φ−1 (y) will have non-empty
intersection, but this is absurd. It is clear that the closed subset φ−1 (y)
must at least contain one closed orbit. This completes the proof.
¤
This proposition implies that the quotient Y does not in general parameterize the G-orbits on X; instead it parameterizes the closed G-orbits on
X.
Remark 2.3.7. Also, it implies that, topologically, the quotient Y is obtained from X by modulo a new equivalence relation:
x ∼ x0 ⇐⇒ G · x ∩ G · x0 6= ∅.
Exercises
1. Let G = C∗ act on P2 by
λ · [x : y : z] = [λx : λ−1 y : z]
as in the introduction. For each of the open subsets below
U− 1 = P2 \ {Y = 0},
2
U 1 = P2 \ {X = 0},
2
U0 = P2 \ {[1 : 0 : 0], [0 : 1 : 0]},
show that the map
1 1
Ui → P1 , i = − , , 0
2 2
[x : y : z] → [xy : z]
defines a categorical quotient.
2. Let X be the variety in Exercise 3 of §2.2. Define φ : X → A1 by
φ(αx + βy, F4 (x, y)) = F4 (−β, α).
17
Show that this is a geometric quotient of X by SL(2).
3. Let G be a reductive algebraic group acting on a normal algebraic variety X. Then the categorical quotient X//G, if exists, is also a normal
algebraic variety.
4. Let G = GL(n) and Mn be the affine space of all n × n matrices. Let G
act on Mn by conjugation, g · x = gxg −1 . For each g ∈ GL(n), consider the
characteristic polynomial
det(g − tIn ) = (−t)n + c1 (g)(−t)n−1 + · · · + cn (g).
Define
c : Mn → Ank
by the formula
c(g) = (c1 (g), · · · , cn (g)).
Show that c : Mn →
is a categorical quotient. Is it geometric? (Hint: it
suffices to check that O(Mn )G = k[c1 , · · · , cn ]. For this, use the fact that any
symmetric polynomial is a polynomial in c1 , · · · , cn .)
Ank
5. Let G be a finite group acting on a quasi-projective scheme X. Show
that the geometric quotient X/G always exists. (The assumption that X
is quasi-projective is important here. Or, one needs to assume that every
orbit is contained in a G-invariant open affine subset. Without this assumption, X/G may not exist as a scheme but is only an algebraic space.
There are such examples even for G = Z2 .)
18
2.4. The Hilbert-Nagata theorem.
Lemma 2.2.1 motivates the following algebraic definition.
Definition 2.4.1. Let G be an algebraic group and R a k-algebra. A rational
action of G on R is a map
R × G → R, , (f, g) → f g
such that
0
0
(1) f gg = (f g )g and f e = f for all f ∈ R and g, g 0 ∈ G;
(2) the map f → f g is a k-algebra automorphsim of R for all g ∈ G;
(3) every element of R is contained in a finite-dimensional invariant
subspace on which G acts by a rational representation.
By Lemma 2.2.1, if G acts on a scheme X, then it acts on O(X) rationally.
Theorem 2.4.2. (Hilbert-Nagata) Let a reductive group G act rationally on a
finitely generated algebra A. Then the subalgebra AG of invariants is also finitely
generated.
Recall that an algebra A is finitely generated if there is a finite set of
elements f1 , · · · , fN of A such that any f ∈ A can be expressed as a polynomial in f1 , · · · , fN with coefficients in k.
We will prove this theorem over a field of charateristics 0, referring the
general case to [73] or [74] (and the references therein). Over a field of
charateristics 0, we will use the fact that a reductive group is necessarily
linearly reductive.
We will first prove
Lemma 2.4.3. Let G be a reductive group acting linearly on k[z1 , . . . , zn ] where
the characteristic of k is zero. Then k[z1 , . . . , zn ]G is finitely generated.
Proof. A = k[z1 , . . . , zn ] is naturally graded
A = ⊕d≥0 Ad .
Hence we have
AG = ⊕d≥0 AG
d.
Since any linear representation of G is completely reducible over charateristics 0, we have a projection operator
r : A → AG
which makes A an AG -module, that is, r(ab) = ar(b) for any a ∈ AG and
b ∈ A.
Now consider the ideal I in A generated by homogeneous polynomials
of positive degree from AG . By Hilbert Basis Theorem ([50], page 417), I
is generated by a finite set of elements f1 , . . . , fN . We may assume that
19
each fi is a homogeneous polynomial of some degree di from AG
di . For any
G
f ∈ Ad we can write
X
ai fi , ai ∈ Ad−di .
f=
i
Then we have
f = r(f ) =
X
r(ai )fi .
i
By using induction on the degree of f we may assume that the r(ai ) are all
polynomials in the fi ’s, hence f is a polynomial in the fi ’s as well.
¤
Lemma 2.4.4. Let a reductive group G act rationally on a finitely generated algebra R over an arbitrary field k, leaving an ideal J invariant. For any 0 6= f ∈
(R/J)G there exists d > 0 such that f d ∈ RG /(J ∩RG ). If G is linearly reductive
(e.g., when the characteristic of k is zero) then d can be chosen to be 1.
Proof. Let h be a representative in R whose image in R/J is f . Let M be
the finite-dimensional2 subspace of R spanned by {hg : g ∈ G}. Write
N = M ∩ J. By hypothesis, h ∈
/ N , but hg − h ∈ N for all g ∈ G because the
image of h in R/J is f and f is invariant. It follows that dim M = dim N +1
and any element of M can be written uniquely as
ah + h0
where a ∈ k and h0 ∈ N . Define a linear map ` : M → k by
`(ah + h0 ) = a,
and one can check that this is G-invariant. Choose a basis {h2 , . . . , hr } of
N ; then {h, h2 , . . . , hr } is a basis of M , and we can identity the dual space
M ∗ with k r by means of the dual basis of M ∗ . With this identification, `
becomes the element (1, 0, . . . , 0) in k r . Note that ` is G-invariant for the
linear action of G on k r . Since G is (geometrically) reductive, there exists
an invariant homogeneous polynomial F ∈ k[z1 , . . . , zn ] of degree d ≥ 1
such that F (`) 6= 0. This implies that the coefficient of z1d in F is non-zero;
We may assume that this coefficient is 1. Finally consider the k-algebra
homomorphism
k[z1 , . . . , zr ] → R
given by
z1 → h, , zi → hi (i ≥ 2).
One checks that this homomorphism commutes with the action of G. It
follows that the image h0 of F in R belongs to RG . Clearly hd −h0 belongs to
the ideal generated by h2 , . . . , hr , which is contained in J. This completes
the proof.
¤
2
The reason that M must be of finite dimension is the same as the one in Lemma 2.2.1.
20
Corollary 2.4.5. Let the characteristic of k is zero. Assume that a reductive
group G acts rationally on a finitely generated k-algebra R. Let J be an ideal in
R, invariant under G. Then (R/J)G = RG /(J ∩ RG ).
Proof. This is because G is linearly reductive and in the above lemma we
can choose d = 1.
¤
Proof of Theorem 2.4.2 over a field of characteristic 0.
Proof. We can assume that A = R/J where R is a polynomial algebra on
which G acts linearly and J is G-invariant ideal. To achieve it, we choose a
set of generators for A and then consider the linear action of G on the finitedimensional vector space V spanned by the translates of the generators.
Choose a basis h1 , . . . , hn for V , then the surjection
R = k[z1 , . . . , zn ] → A, zi → hi
is a surjective G-equivariant homomorphism. Let J be the kernel of this
map. Then A = R/J. By the above corollary, AG = RG /(J ∩ RG ). By
Lemma 2.4.3, RG is finitely generated, hence so is AG .
¤
Exercises
1. Let G be a reductive algebraic group acting algebraically on a normal
finitely generated k-algebra A. Then AG is also a normal finitely generated
k-algebra. (Recall that a normal ring is a domain integrally closed in its
field of fractions. The reader should compare this problem with Exercise 1
in §2.3.)
2. Let φ : X → Y is a good categorical quotient by the action of a reductive
group G. Show that for any open subset U ⊂ Y , the restriction morphism
φ−1 (U ) → U is also a categorical quotient with respect to the induced Gaction.
21
2.5. Quotients of affine schemes.
Any scheme is locally an affine scheme. The idea of GIT is to go from
local to global3. That is, we try to cover a G-scheme X with invariant affine
open subsets, for each affine open subset we take an affine quotient (see the
theorem below) and then glue these resulting affine quotients together to
form a desired quotient X//G. So, we consider quotients of affine schemes
first.
Theorem 2.5.1. Let a reductive group G act on an affine scheme X. Then O(X)G
is finitely generated and the natural surjection
φ : X = Spec(O(X)) → Y = Spec(O(X)G )
induced by the inclusion O(X)G ,→ O(X) is a good categorical quotient.
Proof. By the Hilbert-Nagata theorem, Theorem 2.4.2, O(X)G is finitely
generated. Clearly that φ : X = Spec(O(X)) → Y = Spec(O(X)G ) is a
surjective G-morphism with the trivial G-action on Y . We will now check
the three conditions of Definition 2.3.3.
(1). In fact, it suffices to prove the statement for any open subset of the
form U = Yf , f ∈ O(Y ) = O(X)G . In this case, φ−1 (U ) = Xf . So the
statement is simply “taking invariants commutes with localization”
(O(X)G )f = (O(X)f )G
for any f ∈ O(X)G , which is easy to check.
(3). For any two disjoint invariant closed subsets W0 and W1 of X, by
Lemma 2.2.2 there exists f ∈ O(X)G such that
f (W0 ) = 0, f (W1 ) = 1.
Regarding f as an element of O(Y ), we get
f (φ(W0 )) = 0, f (φ(W1 )) = 1.
Hence
φ(W0 ) ∩ φ(W1 ) = ∅.
This proves (3).
(2). If y ∈ φ(W ) \ φ(W ), we let W0 = φ−1 (y) and W1 = W and apply the
proof of (3), we get a contradiction.
¤
This allows us to define the GIT quotient of Spec(A) by the group G to
be
Spec(A)//G = Spec(AG ).
3
In contrast, when we construct a quotient as an algebraic space, the method is complete global. This will be discussed in detail in later sections
22
Example 2.5.2. Let X = C2 , G = C∗ with the action
G×X →X
given by
t · (z1 , z2 ) = (tz1 , t−1 z2 ).
Any orbit corresponds to an equation
z1 z2 = c
where c is a complex constant except when c = 0 in which case the equation
z1 z2 = 0
splits into three distinct orbits
{(z1 , 0) : z1 6= 0}, {(0, z2 ) : z2 6= 0}, {(0, 0)}.
Thus if we take the ordinary orbit space X/G, we obtain a complex line
with three origins. In particular, it is non-Hausdorff.
Categorical GIT quotients are very different. The algebra of regular
functions on X = C2 is O(X) = C[z1 , z2 ]. Consider the action of C∗ on
O(X) as before. That is, for any h ∈ O(X) and t ∈ C∗ , define
ht (z1 , z2 ) = h(tz1 , t−1 z2 ).
It is easy to check that
O(X)G = C[z1 z2 ].
That is, O(X)G ∼
= C[z] by setting z = z1 z2 , hence
X//G = Spec O(X)G = C.
In particular, it is separated (i.e., Hausdorff). In other words, instead taking just orbits space, GIT suggests that we identify the three orbits defined
by the single equation z1 z2 = 0. One should compare this with Proposition
2.3.6 (1) which suggests exactly the same thing.
Exercises
1. We can twist the action of Example 2.5.2 by characters of C∗ to obtain
different actions on O(X) as follows. That is, for any h ∈ O(X) and t ∈ C∗ ,
define linear actions of C∗ on O(X) by
ht (z1 , z2 ) = t−1 h(tz1 , t−1 z2 ).
and
ht · (z1 , z2 ) = th(tz1 , t−1 z2 ),
respectively. Compute the subspace Γ(X, O(X))G of the invariants for
each of the above actions. (Note that here the set of invariants is only a
linear subspace but not a subring. Nevertheless, this example suggests, as
23
in the following exercise, that even for an action on the affine space, there
are potentially many different quotients.)
2. Given a linear action σ of G on O(X), define an open subset Uσ ⊂ X by
Uσ = {x ∈ X|∃f ∈ Γ(X, O(X))G , f (x) 6= 0}.
For each linear action of G on O(X) in Example 2.5.2 and the previous
exercise, compute Uσ and, if exists, its categorical quotient.
24
2.6. Linearization and stability.
As mentioned earlier, the idea of GIT is to go from local to global. More
precisely, we will try to cover an invariant open subset U by invariant
affine open subsets such that for each of these affine open subsets we can
take the affine quotient whose existence was shown in the previous section, and, then glue these affine quotients together to form the global quotient “U/G”, as desired.
The choice of a covering of invariant affine open subsets can conveniently done by choosing an equivariant projective embedding of X, or
equivalently (for this purpose), a G-linearization.
Definition 2.6.1. A G-linearization of the action of G on X is a line bundle
π : L → X together with a lifting of the action to L
G × L −−−→ L




y
y
G × X −−−→ X
such that
(1) π(g · v) = g · π(v) for any g ∈ G and v ∈ L;
(2) given any x ∈ X, g ∈ G, v ∈ Lx the induced map
Lx → Lg·x , v → g · v
is linear.
L equipped with a linearization is called a linearized line bundle or an
equivariant polarization. Clearly, if L is linearized, it induces a linearization on L⊗n .
Remark 2.6.2. If we assume that L is very ample and defines an embedding
X ,→ P(H 0 (X, L)∗ )
where H 0 (X, L)∗ is the linear dual of the space of sections of L, then a
linearization on L is equivalent to saying that G acts linearly on the homogeneous coordinates of X in P(H 0 (X, L)∗ ). In this case, sections of Ld
correspond to homogeneous polynomials of degree d.
Any linearization on a line bundle L induces a G-action on Γ(X, L) as
follows: for any s ∈ Γ(X, L) and g ∈ G
g
s(x) = g −1 · s(g · x).
We say a section s is G-invariant if g s = s for all g ∈ G. This simply means
that
s(g · x) = g · s(x)
for all g ∈ G and x ∈ X.
25
Example 2.6.3. If L is the trivial line bundle O(X) with the trivial G-action,
then we have
Γ(X, L) = O(X) and Γ(X, L)G = O(X)G .
In this case, O(X)G is an algebra. Even if L is the trivial line bundle O(X),
if the action on L is not trivial, we may not have Γ(X, L)G = O(X)G , and
the set of invariants Γ(X, O(X))G is only a linear subspace of O(X). The
different actions on O(X) that we considered in Exercise 2.5 are examples
of different linearizations.
Remark 2.6.4. Now is the time to make a convention. When we write
O(X)G , the corresponding linearization is understood as the trivial line
bundle with the trivial action. For any other action on the trivial line bundle O(X), we will write Γ(X, O(X))G . O(X)G is a subalgebra. Γ(X, O(X))G
is in general only a linear subspace.
For any given quasi-projective G- scheme X, a linearized ample line
bundle L provides a natural way to produce a G-invariant open subset
covered by invariant affine open subsets. Indeed, take any invariant section s of L⊗n , then the open subset
Xs = {x ∈ X|s(x) 6= 0}
is affine (because L is ample) and G-invariant. The union of all these open
subsets is an invariant open subset with the desired properties.
This motivates the definition of stability.
Definition 2.6.5. Let X be a quasi-projective G-scheme and L a linearized
line bundle.
(1) x ∈ X is semistable with respect to L if s(x) 6= 0 for some s ∈
Γ(X, L⊗n )G for some n and Xs is affine; the set of semistale points
with respect to L is denoted by X ss (L);
(2) x ∈ X ss (L) is stable with respect to L if G · x is closed in X ss (L) and
Gx is finite; the set of stable points is denoted by X s (L);
(3) x ∈ X ss (L) is called polystable4 with respect to L if x ∈ X ss (L) and
G · x is closed in X ss (L); the set of polystable points is denoted by
X ps (L)
(4) a non-semistable point is called unstable (this is somewhat unfortunate but is completely standard); the set of unstable points is denoted by X us (L).
4
In some moduli problems, G · x is closed in X ss (L) often means that the geometric
object represented by the point x splits as a direct sum of smaller stable objects. For
example, this is the case when x represents a vector bundle where the term “polystable”
is standard.
26
We observe immediately that replacing L by Lm does not change stability.
When L is ample, the requirement that Xs is affine is automatically satisfied, and thus is redundant. In any case, for any linearized line bundle
L, X ss (L), when nonempty, is covered by invariant affine open subsets of
the form Xs .
The following lemma shows that a linearized line bundle L, when restricted to X ss (L), must be ample.
Lemma 2.6.6. (A criterion for ampleness) Let L be a line bundle over a scheme
X. Assume that for every x ∈ X there exists a section t ∈ Γ(X, Ln ) for some n
(depending on x) with t(x) 6= 0 and Xt is affine. Then L is ample.
Proof. X can be covered by {Xt1 , . . . , Xtr } for some sections t1 , . . . , tr of
Ln1 , . . . , Lnr , respectively, Let n = lcm{n1 , . . . , nr }, we may assume that all
ti ∈ Γ(X, Ln ).
For any coherent sheaf F on X, F |Xti is generated by finitely many
global sections since Xti is affine. Hence for mi >> 0, those sections lie
in
Γ(X, F ⊗ OX (mi V(ti )))
where V(ti ) is the vanishing locus of ti and F ⊗ Lmn is globally generated
over Xti for all m ≥ mi . Let m0 = max{mi }, we see that F ⊗ Lnm is
generated by global sections for all m ≥ m0 . Hence Ln is ample, then so is
L as well.
¤
Corollary 2.6.7. Assume that X ss (L) is not empty. Then Tthe restriction L|X ss (L)
is ample.
Hence we may, without loss of much generality, only concentrate on
linearized ample line bundles.
Remark 2.6.8. We should point out here that there is lack of literature on
non-ample linearizations. They do in general produce quotients that are
not coming from ample linearized line bundles.
Example 2.6.9. Consider the action
C∗ × Cp+q → Cp+q
defined by
λ · (z1 , · · · , zp , w1 , · · · , wq ) = (λz1 , · · · , λzp , λ−1 w1 , · · · , λ−1 wq ).
Every line bundle on Cp+q is trivial. So, let L = Cp+q × C be the trivial line
bundle. C∗ acts on the fiber C by a character χ ∈ Z. That is, λ · v = λχ v. Let
Lχ be the corresponding linearized line bundle. Note that a section of L is
simply a function on Cp+q .
There are three cases depending on the sign of the integer χ.
27
Case χ = 0. Since the action is trivial, any constant function is C∗ invariant, hence
X ss (L0 ) = Cp+q .
Case χ = 1. Let s : Cp+q → C be a section. s is invariant if
s(λz1 , · · · , λzp , λ−1 w1 , · · · , λ−1 wq ) = λs(z1 , · · · , zp , w1 , · · · , wq )
for all λ ∈ C∗ . This implies that s must be of the form
s = l(z1 , · · · , zp )f (· · · zi wj · · · )
where l is linear and homogeneous in (z1 , · · · , zp ) and f is a polynomial
function in (· · · zi wj · · · ). It follows that
X ss (L1 ) = {z1 , · · · , zp , w1 , · · · , wq ) ∈ Cp+q |zi 6= 0 for some i}.
Case χ = −1. Let s : Cp+q → C be a section. s is invariant if
s(λz1 , · · · , λzp , λ−1 w1 , · · · , λ−1 wq ) = λ−1 s(z1 , · · · , zp , w1 , · · · , wq )
for all λ ∈ C∗ . This implies that s is of the form
s = l(w1 , · · · , wq )f (· · · zi wj · · · )
where l is linear and homogeneous in (w1 , · · · , wq ) and f is a polynomial
function in (· · · zi wj · · · ). It follows that
X ss (L1 ) = {z1 , · · · , zp , w1 , · · · , wq ) ∈ Cp+q |wj 6= 0 for some j}.
A general result from the next section will show that the good quotient
exists for any of these three open subsets. But, it can also be done directly.
Remark 2.6.10. We note here that for a general scheme, it is in general
highly non-trivial to find invariant affine open covers. In fact, it is not true
that any G-scheme X can be covered by invariant affine open subsets even
if we assume that a categorical quotient of X by G exists. (In other words,
what GIT does is to give a sufficient but not necessarily necessary way.)
Example 2.6.11. Let Xn,m denote the space of hypersurfaces of degree m in
Pn . The group PGLn+1 acts obviously on Xn,m . Since Xn,m is a projective
space and PGLn+1 has trivial character, stability has only one sense for this
action. Surprisingly (maybe not), it is very hard to determine the whole
semistable locus using invariant sections, except for reasonably small m
and n. But, we do have an easy partial result in general.
Claim: If a hypersurface H with degree m ≥ 3 is nonsingular, then it is
stable.
To prove this, observe that for any hypersurface H defined by a homogeneous form F of degree m, there is a homogeneous polynomial ∆ in the
coefficients of F , invariant under the action of PGLn+1 , called the discriminant of F (or H), such that H is nonsingular if and only if ∆ 6= 0. This
shows that H is semistable in Xn,m . Since m ≥ 3, the isotropy subgroup of
28
PGLn+1 at H is finite. (This, for example, follows from the nonexistence of
global vector fields on such hypersurfaces.) Therefore, H is stable.
For a description of the whole semistable locus, see Exercise 3.2. (3).
Exercises
1. Let H be a closed reductive subgroup of G. Then prove that
(1) any G-linearization L induces an H-linearization LH ;
(2) any G-stable (resp. semistable) point with respect to L is also H
-stable (resp. semistable) point with respect to LH .
2. Consider the action
C∗ × Cp+q → Cp+q
as in Example 2.6.9. Show that for any character χ, X ss (Lχ ) coincides with
one of X ss (L−1 ), X ss (L0 ) and X ss (L1 ).
3. Let a0 , · · · , an be a sequence of positive integers. Let C∗ act on Cn+1 by
t · (z0 , · · · , zn ) = (ta0 z0 , · · · , tan zn ).
Describe all the linearizations, their semistable points and their categorical
quotients.
4. Let x ∈ Xs ⊂ X ss (L) for some s ∈ Γ(X, Lm ). Show that G · x is closed in
X ss (L) if and only if it is closed in Xs .
5. Show that x is stable with respect to a linearization L if and only if there
is an invariant section s with s(x) 6= 0 such that Xs is affine, the action on
Xs is closed (i.e., all orbits are closed), and Gx is finite.
29
2.7. Existence of GIT quotients.
Theorem 2.7.1. Let G act on a scheme X and L a linearized line bundle with
X ss (L) 6= ∅. Then
(1) a good quotient X ss (L) → X ss (L)//G exists;
(2) the geometric quotient X s (L) → X s (L)/G exists and is embedded in
X ss (L)//G as an open subset.
(3) the quotient morphism X ss (L) → X ss (L)//G is affine.
Proof. (1). X ss (L) is covered by affine open subsets of the form U = Xs
where s is an invariant section of L⊗n for some n > 0. Since X is Noetherian, there are finitely many such sections s1 , . . . , sk such that
X ss (L) = ∪i Ui
where Ui = Xsi . By the Hilbert-Nagata theorem, the categorical quotient
φi : Ui → Ui //G = Vi
exists. Now let us find the patching maps for the affine open subsets {Vi }.
Again since X is Noetherian, we may assume that there is a large integer N
such that s1 , . . . , sk are elements of H 0 (X, L⊗N ). sj /si is a regular invariant
function5 of Γ(Ui , OX ) and hence is induced by a function σij of Γ(Vi , OVi ).
Define
Vij = Vi \ {y : σij (y) = 0}.
Then
−1
φ−1
i (Vij ) = Ui ∩ Uj = φj (Vji ).
That is, both Vij and Vji are categorical quotients of Ui ∩ Uj . Therefore there
is a unique isomorphism
∼
=
ψij : Vij −−−→ Vji
such that the following diagram
=
Ui ∩ Uj −−−→ Ui ∩ Uj




φj y
φi y
Vij
ψij
−−−→
Vji .
commutes. It follows that {ψij } patch {Vi } together to form a prescheme
(not necessarily separated) containing {Vi } as affine open subsets.
Next notice that the functions {σij } form a Cech 1-cocycle for the covering {Vi } of Y and in the sheaf OY∗ . Therefore these functions, as transition
5
Here that assumption that the action of G on L is linear is used to guarantee that sj /si
is invariant.
30
functions, define an invertible sheaf M on Y . Moreover, by the construction, it is clear that LN = φ∗ (M ) since φ∗ (σij ) = sj /si form transition functions for LN .
To show that M is ample, we will apply Lemma 2.6.6. Observe first that
{σij }i (for a fixed j) is a collection of functions such that on Vi ∩ Vj
σjk = σik · σji .
Therefore it defines a section tj of M . To see this, note that tj |Vi = σij
and for two open subsets Vi1 and Vi2 , tj |Vi1 and tj |Vi2 differ by a transition
function on the overlap Vi1 ∩ Vi2 . One check by definition that
sj = φ∗ (tj ).
Hence
sj (x) = 0 if and only if tj (φ(x)) = 0.
That is,
Vj = Ytj = Y \ {y : tj (y) = 0}.
Since {Vj = Ytj } is an affine covering of Y , by Lemma 2.6.6 (cf. [30], page
155), M is ample. This implies that Y is quasi-projective, in particular, a
scheme.
(2) can be easily checked.
(3) follows from the proof of (1).
¤
Remark 2.7.2. Note here that the linear action of G on L is used to show
that sj /si is a function on Ui . It seems to be an interesting question whether
the linear action is only a sufficient condition but not necessary.
L
Theorem 2.7.3. X ss (L)//G = Proj( n≥0 H 0 (X, L⊗n )G ).
Proof. Replacing L by Ld for a large d, we may assume that the ring
M
RG =
H 0 (X, L⊗n )G
n≥0
is generated by elements s0 , . . . , sn of degree 1. Then
Y = Proj(RG )
as a subscheme of Pn is defined by the homogeneous ideal of the kernel of
the surjection
C[T0 , . . . , Tn ] → RG , Ti → si .
Notice that the affine open subsets
{Ui = Xsi }
cover X ss (L). On the other hand,
Yi = Y ∩ {Ti 6= 0}
31
form an affine open cover of Y with the property that
OYi = OUGi .
Thus the quotient maps
Ui → Y i
define a morphism
X ss → Y
which verifies the definition of a categorical quotient whose uniqueness
implies the theorem.
¤
We have mentioned earlier, in the introduction and through the exercises of the previous sections, that even for an action on affine space, we
can have (potentially) different quotients. We now demonstrate this by
examining an easy example.
Example 2.7.4. Let X = C3 and G = C∗ with the action given by
G×X →X
t · (z1 , z2 , w1 ) = (tz1 , tz2 .t−1 w1 ).
This is the special case of Example 2.6.9 when p = 2 and q = 1. We will
leave the general case to the reader as an exercise.
First observe that G-orbits are parameterized by equations
z1 w1 = c1 , , z2 w1 = c2
except when (c1 , c2 ) = (0, 0) in which case the equation splits into several
orbits with {(0, 0)} as the unique closed orbit.
Recall from Example 2.6.9 that we have X ss (L0 ) = X. By identifying the
orbits whose closures meet non-trivially (see Remark 2.3.7), we can check
that the equivalence classes are parameterized by equations
z1 w1 = c1 , , z2 w1 = c2 ,
that is, by a pair of complex numbers (c1 , c2 ). This implies that the morphism
X ss (L0 ) → C2
(z1 , z2 , w1 ) → (z1 w1 , z2 w1 )
is a categorical quotient and hence X ss (L0 )//G = C2 .
Now consider the different open subset
X ss (L1 ) = {(z1 , z2 , w1 ) : z1 or z2 6= 0}.
Every orbit in this open subset is closed in X ss (L1 ). Hence, this is a geometric quotient. One checks that the inclusion X ss (L1 ) ⊂ X ss (L0 ) induces
a surjective morphism
p : X ss (L1 )/G → X ss (L0 )//G
32
by sending a point
G · (z1 , z2 , w1 ) ∈ U1 /G
to the point
(z1 w1 , z2 w1 ) = (c1 , c2 ) ∈ U0 //G.
It can verified directly that p is isomorphic over C2 \ {0} but
p−1 (0, 0) = {C∗ · (z1 , z2 , 0) : z1 or z2 6= 0} = P1 .
That is, X ss (L1 )/G is the blowup of X ss (L0 )//G along the origin.
There is yet the third quotient X ss (L−1 )/G which is, as one can easily
verify, isomorphic to X ss (L0 )//G.
These quotients, although distinct, are nevertheless closely related. We
will come back to this in a general setting later.
Exercises
1. Describe the quotients of the open subsets in Example 2.6.9 and study
how exactly these quotients are related.
2. Let C∗ acts on Pp+q−1 by
t · [t−a1 z1 : · · · t−ap zp : tb1 w1 : · · · tbq wq ]
where a = (a1 , · · · , ap ), b = (b1 , · · · , bq ) are sequences of positive integers.
Show that
(Pp+q−1 )ss = (Pp+q−1 )s = {[z : w|z 6= 0, w 6= 0]
and the quotient is
Pp−1 (a) × Pq−1 (b).
3. Let E be a vector bundle of rank k. A linearization on E is a commutative diagram
G × L −−−→ E




y
y
G × X −−−→ X
such that the action on every fiber is linear, that is, in Definition 2.6.1, we
simply replace L by E and maintain everything else.
(1) x ∈ X is semistable with respect to E if there are invariant sections
s1 , · · · , sk ∈ Γ(X, E ⊗n )G for some n such that det(s1 (x), · · · , sk (x)) 6=
0 and Xs1 ,··· ,sk is affine. Here we write each si (x) as a column vector
and Xs1 ,··· ,sk is the set of points such that det(s1 , · · · , sk ) 6= 0. The
set of semistale points with respect to E is denoted by X ss (E);
(2) x ∈ X ss (E) is stable with respect to E if G · x is closed in X ss (E)
and Gx is finite; the set of stable points is denoted by X s (E);
33
(3) x ∈ X ss (E) is called polystable with respect to E if x ∈ X ss (E) \
X s (E) and G · x is closed in X ss (E);
(4) a non-semistable point is called unstable. The set of unstable points
is denoted by X us (E).
When E is ample, that is, the line bundle det(E) is ample, the requirement
that Xs1 ,··· ,sk (x) is affine is automatically satisfied, and thus is redundant.
Prove that X ss (E) admits a good projective quotient X ss (E)//G. (Hint:
Prove that X ss (E) = X ss (det(E)). Hence, replacing line bundles by vector
bundles does not produce more quotients.)
Problems
4. In the proof of Theorem 2.7.1, the assumption that the actions of G on the
fibers of L are linear is used to guarantee that sj /si is an invariant function
(see the footnote there). If we do not assume that the action of G on L is
linear, but instead, we replace it with that all sj /si are invariant functions,
while maintaining all other assumptions, does this implies that the action
of G on L must be linear? Prove this or construct a counterexample.
5. Consider the complete flag variety FLn = {V1 ⊂ V2 , ⊂ · · · , ⊂ Vn−1 ⊂
Cn }. Choose a coordinate system and G = (C∗ )n /∆ acts on FLn by multiplying the coordinates. Here ∆ stands for the diagonal subgroup which
acts trivially on FLn . Define the open subset U by requiring that a flag
V1 ⊂ V2 , ⊂ · · · , ⊂ Vn−1 belongs to U if Vn−1 is in general linear position
with respect to the coordinate hyperplanes. Prove that the geometric quotient U/G exists and can be identified with the flag variety FLn−1 . Since
FLn−1 is projective, we know that there is ample linearization over U such
that U ss = U s = U . For n ≥ 4, is there an ample linearized ample line bun0
dle L over FLn such that U = FLss
n ? (One can also define U by requiring
that a flag V1 ⊂ V2 , ⊂ · · · , ⊂ Vn−1 belongs to U 0 if V1 is in general linear position with respect to the coordinate hyperplanes and ask the same
questions.)
34
3. C RITERIA FOR S TABILITY
Having the foundational Theorem 2.7.1, the next step is to provide some
feasible tests for stability, as the stability of a point is hard to check using
the definitions.
The most effective algebraic test is the Hilbert-Mumford numerical criterion. The most useful geometric criterion is via moment map.
3.1. A geometric criterion.
Stability depends on linearization. As far as stability is concerned, a linearized (very) ample line bundle is equivalent to an equivariant projective
embedding, which we now explain.
Let L be a linearized ample line bundle. Replace L by a tensor power if
necessary, we may assume that L is very ample. Hence a choice of basis of
Γ(X, L) will define an embedding
X ,→ Pn = P(H 0 (X, L)∗ )
such that G acts linearly on the homogeneous coordinates of X in Pn . Under these homogeneous coordinates, Γ(X, Lm ) consists of degree m homogeneous polynomial functions on X. This is the setup which we will
repeatedly use in the rest of this chapter.
Set V = H 0 (X, L)∗ and
π : V \ {0} → Pn
be the projection. For any x ∈ X, let x̂ be any lifting of x in V \ {0}.
Theorem 3.1.1. (A geometric criterion.) We have
(1)
(2)
(3)
(4)
x ∈ X ss (L) if and only if 0 ∈
/ G · x̂;
x is polystable if and only if G · x̂ is closed in V
x ∈ X s (L) if and only if G · x̂ is closed in V and Gx̂ is finite;
x ∈ X us (L) if and only if 0 ∈ G · x̂.
Proof. (1). We prove the sufficiency first. If 0 ∈
/ G · x̂, then {0} and G · x̂ are
two disjoint invariant closed subset of V . Since G is reductive, we can find
a G-invariant polynomial F separating the two, e.g., such that F (x̂) = 1
and F (0) = 0. Write F as the sum of its homogeneous parts
F = F0 + F1 + . . . + Fr .
Hence F0 = 0 but Fi (x̂) 6= 0 for some 0 < i ≤ r. But then Fi defines a
section s ∈ Γ(X, L⊗i )G with s(x) 6= 0. This implies that x ∈ X ss (L).
Conversely if s ∈ Γ(X, L⊗i )G with s(x) 6= 0 and F is the corresponding
G-invariant homogeneous polynomial of degree i, then F (x̂) 6= 0. For any
point v ∈ G · x̂, F (v) = F (x̂) 6= 0. Hence 0 ∈
/ G · x̂.
35
(2). If G · x̂ is closed, then 0 ∈
/ G · x̂. Thus, x is semistable. Hence by
the proof of (1), we can select a G-invariant homogeneous polynomial F
of degree d such that
G · x̂ ⊂ Z = {F = constant 6= 0} ⊂ V \ {0}.
Let s be the G-invariant section corresponding to F . Then we have
γ
ϕx : G −−−→ G · x̂ ,→ Z −−−→ Xs .
By the construction of Z, γ is surjective and finite, hence proper, in particular closed. Hence, G · x is closed in Xs , and therefore closed in X ss (L) (see
Exercise 2.6.4). Thus, x is polystable.
Conversely, if x is polystable, but G · x̂ is not closed. Then G · x̂ contains
a lower dimension orbit G · ŷ. Because x is semistable, 0 ∈
/ G · x̂, hence
0∈
/ G · ŷ. Thus, y is semistable. This contradicts with that x is polystable.
(3) immediately follows from (2).
(4) is the opposite of (1).
¤
Extending some arguments in the above proof, we can obtain a general
criterion for stable points in terms of the properness of the map ϕx and ϕx̂ .
To this end, we need two lemmas.
Lemma 3.1.2. Let a reductive algebraic group G act on a scheme X and L is a
linearization. Let x ∈ X ss (L). Then ϕx is proper if and only if ϕx̂ is proper.
Proof. Since x ∈ X ss (L), we obtain that {0} and G · x̂ are two disjoint invariant closed subset of V . Using the notations from the proof of Theorem
3.1.1, the map ϕx̂ can be factorized as
ϕx̂ : G −−−→ G · x̂ ,→ Z ,→ V,
and the map ϕx can be factorized as
γ
ϕx : G −−−→ G · x̂ ,→ Z −−−→ Xs .
Observe that γ is finite, surjective, and hence proper. Then the properness
of the two maps are equivalent because, when either of them is proper, Gx̂
must be finite and G · x̂ must be closed.
¤
Lemma 3.1.3. Let σ : G × W → W be an algebraic action. Then
ϕx : G → W
g →g·x
is proper if and only if G · x is closed in W and Gx is finite.
Proof. ϕx factors over the morphism G → G/Gx as
∼
=
G → G/Gx −−−→ G · x ,→ W.
36
Since G is affine, the properness of the morphism G → G/Gx is equivalent
to the properness of Gx which is in turn equivalent to its finiteness. The
properness of the inclusion G · x ,→ W is same as closeness. The lemma
follows.
¤
In this lemma, replacing W by X ss (L), we obtain
Proposition 3.1.4. (A general criterion for stable points). Let a reductive
algebraic group G act on a scheme X and L is a linearization. Then x ∈ X ss (L)
is stable if and only if ϕx is proper if and only if ϕx̂ is proper.
Exercises
1. Let the notation be as in Theorem 3.1.1. Define
M
N (X, L) = {x̂ ∈ V |s(x̂) = 0, ∀s ∈
Γ(X, Lm )G }.
m≥0
Here we identify Γ(X, Lm ) with degree m homogeneous polynomial functions on X ⊂ P(V ). Show that N (X, L) is an affine cone.
2. Let x ∈ X ss (L) and R be a reductive subgroup of G. Show that R fixes x
if and only if R fixes any (or all) x̂. (Hint: Since R is reductive, it fixes x̂ if
and only if every one-parameter subgroup does. Then use the geometric
criterion for semistability.)
3. Let S be any closed subgroup of G. G/S is affine if and only if S is
reductive. Use this to show that if x ∈ X ps (L), then both Gx and Gx̂ are
reductive.
4. Prove that the morphism
Ψ : G × Xs → Xs × Xs
(g, x) → (g · x, x)
is proper.
37
3.2. The Hilbert-Mumford numerical criterion.
Checking stability is hard in general. The following Hilbert-Mumford
numerical criterion makes it more accessible (but may still be difficult in
practice).
To motivate the numerical criterion, let us examine the case of semistable
points.
Let the setup be the same as in §3.1. By Theorem 3.1.1 (the geometric
criterion for stability), a point x is semistable only if 0 ∈
/ G · x̂ only if for
any one-parameter subgroup λ
lim λ(t) · x̂ 6= 0.
t→0
Write
λ(t) · x̂ = (tw0 x̂0 , . . . , twn x̂n ).
Then the above is equivalent to requiring that some weights with nontrivial coordinates are non-positive. That is,
min{wi : x̂i 6= 0} ≤ 0.
Definition 3.2.1. µL (x, λ) = min{wi : x̂i 6= 0}.
The above serves as the (working) definition of the numerical function
µL (x, λ) and provides a good intuitive reason for the numerical criterion
(Theorem 3.2.3). Note here that this definition of µL (x, λ) depends on that
L is (very) ample. In general, it can be equivalently defined as follows.
Definition 3.2.2. Let L be any linearized line bundle (not necessarily ample). For any x ∈ X and λ ∈ χ(G)∗ , set y = limt→0 λ(t) · x. Then λ fixes y
and hence acts on the fiber Ly by a character r. Set µL (x, λ) = r.
This gives a coordinate-free definition of µL (x, λ). It is easy to see that
µ (x, λ) = mµL (x, λ). When L is very ample, the above two descriptions
of µL (x, λ) are equivalent. We leave these as exercises for the reader to
verify. We note here that for the rest of this section, we will only rely on
the first description of µL (x, λ).
Lm
Let χ∗ (G) = Hom(Gm,k , G) denote the set of all one parameter subgroups
of G.
Theorem 3.2.3. (The numerical criterion.)
(1) x ∈ X ss (L) if and only if µL (x, λ) ≤ 0 for all λ ∈ χ∗ (G);
(2) x ∈ X s (L) if and only if µL (x, λ) < 0 for all λ ∈ χ∗ (G);
(3) x ∈ X us (L) if and only if µL (x, λ) > 0 for some λ ∈ χ∗ (G).
If µL (x, λ) > 0, then we will say that the one-parameter λ destabilizes x.
A few words before we give the details of the proof.
38
The proof of necessary direction of (1) is already done. The other direction follows from a theorem of D. Birkes which asserts that if G · x specializes to some point, then it does so through the path of the orbit of a
one-parameter subgroup ([7]). The difficult part is the stable case. For
this, we use the properness of the morphism ϕx̂ by the general criterion.
That is, if x is not stable, then ϕx̂ is not proper. We then apply the valuation criterion of properness to the ring R = k[[z]] and its field of fraction
K = k((z)). The goal is to find a one-parameter subgroup that destabilizes
x. To this end, we will need a theorem of Iwahori, which will help us to
single out a one-parameter subgroup λ that destabilizes x, as desired.
The theorem of Iwahori claims the surjectivity of the natural map
ρ : χ∗ (G) → G(k[[z]])\G(k((z)))/G(k[[z]])
where R = k[[z]] is the ring of formal power series with coefficients in k
and K = k((z)) its fraction field, and a one parameter-subgroup λ in χ∗ (G)
is considered as a k((z))-valued point of G by the means of composition
with the canonical map
λ
Spec k((z)) → Spec k[z, z −1 ] = Gm −−−→ G.
Alternatively, one considers λ as a k(z)-point of G and identifies k(z)
with subfield of k((z)) by using the Laurent expansion of rational functions at the origin of A1k . For any λ ∈ χ∗ (G) as an element of G(k[z, z −1 ]),
we will use [λ] to denote the induced image in G(k((z)).
Lemma 3.2.4. (Iwahori’s Theorem [49]) For any reductive algebraic group G,
any element of the set of the double cosets
G(k[[z]])\G(k((z)))/G(k[[z]])
can be represented by a one-parameter subgroup λ. That is,
ρ : χ∗ (G) → G(k[[z]])\G(k((z)))/G(k[[z]])
is surjective.
Proof. We shall only prove for the case when G = SL(n + 1, k) or GL(n +
1, k), referring to the original paper of Iwahori and Matsumato for the general case (see [49]).
Recall that we have set R = k[[z]] and K = k((z)) for simplicity.
A K-point of G is a matrix A with entries in K. We can write it as a
matrix z r B where B ∈ GL(n, R) for some r ≥ 0. As R is a P.I.D, we can
reduce the matrix B to the diagonal form D so that
A = C1 DC2
39
where Ci ∈ G(R) and D is a diagonal matrix with entries z r1 , . . . , z rn . Now
define a one-parameter subgroup of G by
λ(t) = diag(tr1 , . . . , trn ).
Then λ represents the double coset of A ∈ G(K) and hence the statement
of the theorem is verified.
¤
Proof of Theorem 3.2.3, the numerical criterion.
Proof. (1). x is semistable if and only if 0 ∈
/ G · x̂. Since G is reductive, by
a theorem of Birkes ([7]), this is true if and only if for any one-parameter
subgroup λ ∈ χ∗ (G)
lim λ(t) · x̂ 6= 0.
t→0
Write
λ(t) · x̂ = (tw0 x̂0 , . . . , twn x̂n ).
Then the above is equivalent to requiring that some weights with nontrivial coordinates are non-positive. That is,
µL (x, λ) = min{wi : x̂i 6= 0} ≤ 0.
(2). To prove the necessity, assume that x is stable. Then it is semistable.
Hence µL (x, λ) ≤ 0 for all λ ∈ χ∗ (G). Suppose the contrary that there is a
one-parameter subgroup λ such that µL (x, λ) = 0. Again, we write
λ(t) · x̂ = (tw0 x̂0 , . . . , twn x̂n )
and
µL (x, λ) = min{wi : x̂i 6= 0} = 0.
There are two cases.
1. There are some positive wi , in this case, we see that λ(t) · x̂ specializes
to some point outside of the λ-orbit λ · x̂. That is, λ · x̂ is not closed. By
Theorem 3.1.1 (2) (the geometric criterion), x is not λ-stable with respect
to the induced λ-linearization. But, by the definition of the stability, a Gstable point must also be λ-stable (see Exercise 2.6.1), a contradiction.
2. All wi are zero, in this case, λ fixes x̂, again by Theorem 3.1.1 (2), x is
not stable, a contradiction.
Now we prove the sufficiency. Assume that µL (x, λ) < 0 for all λ ∈
χ∗ (G) but x is not stable. We want to produce a one-parameter subgroup
λ0 such that µL (x, λ0 ) ≥ 0 to obtain contradiction. To this end, observe by
Proposition 3.1.4 that the map
ϕx̂ : G → V, g → g · x̂
is not proper. By the valuation criterion of properness, there is a k((z))valued point of G such that it is not a k[[z]]-valued point but it gives an
k[[z]]-valued point of V via the morphism ϕx̂ : G(k((z))) → V (k((z))). Put
40
it differently, there is a k[[z]]-valued point of V such that when it is viewed
as a k((z))-valued point of V , it has a pre-image under
ϕx̂ : G(k((z))) → V (k((z)))
but this pre-image does not come from any k[[z]]-point of G. That is, there
exists an element
g ∈ G(k((z))) \ G(k[[z]])
such that g · x̂ ∈ V (R) = Rn+1 . By Iwahori’s theorem, we can write
g = g1 [λ]g2
where g1 , g2 ∈ G(R) and λ is a one-parameter subgroup viewed as an element of G(K). Let g¯2 be the image of g2 under the “reduction” homomorphism (modulo (z))
G(R) → G(k)
corresponding to the natural homomorphism
X
R → k,
ai z i → a0 .
i
We have
ḡ2−1 g1−1 g = (ḡ2−1 [λ]ḡ2 )ḡ2−1 g2 .
ḡ2−1 [λ]ḡ2 is a K-point of G defined by the one-parameter subgroup λ0 =
ḡ2−1 λḡ2 of G. Choose a basis {e0 , . . . , en } of k n+1 such that the action of λ0 is
diagonalized. Then we may write
λ0 (t) · ei = tri ei , i = 0, . . . , n.
This is equivalent to
[λ0 ] · ei = z ri ei , i = 0, . . . , n.
P
Thus if we write x̂ = i x̂i ei , we get
(ḡ2−1 g1−1 g · x̂)i = ([λ0 ] · (ḡ2−1 g2 · x̂))i = z ri ((ḡ2−1 g2 · x̂))i .
Since g · x̂ ∈ Rn+1 , this implies that
(ḡ2−1 g2 · x̂))i = z −ri (ḡ2−1 g1−1 g · x̂)i ∈ z −ri R.
(∗)
We claim that ri ≥ 0 if x̂i 6= 0. In fact, the element ḡ2−1 g2 is reduced to
the identity modulo (z), hence (ḡ2−1 g2 · x̂))i = x̂i modulo (z). On the other
hand, it is equal to z −ri ai modulo (z) for some ai ∈ R. This implies that
ri ≥ 0 if x̂i 6= 0. This is, µ(x, λ0 ) ≥ 0.
This completes the proof of (2).
(3) is the opposite of (1).
¤
41
Example 3.2.5. Let G be any semi-simple group with Lie algebra g. G acts
on g by adjoint action. Then ξ ∈ g is unstable if and only if adξ is nilpotent.
All other points are strictly semistable.
The characteristic polynomial det(tI − adξ) is G-invariant, hence its coefficients are invariant functions. If ξ is unstable, these functions all vanish
at ξ. Hence adξ is nilpotent.
Conversely, if adξ is nilpotent then {exp(tξ) : t ∈ k} is a unipotent
subgroup of G. Let B be a Borel subgroup such that {exp(tξ) : t ∈ k}
is contained in its unipotent radical Ru (B) and write B = Ru (B)T where
T is some maximal torus of G. Then we have
X
g=t+
gα
α
P
where t = Lie T and α>0 gα = Lie Ru B. Choose an integral point v in the
interior of the positive Weyl chamber in Lie T and let λ = {exp(sv) : s ∈ k}
be the one-parameter subgroup of T generated by this element. We can
write ξ as
X
ξα
ξ=
α>0
where some ξα are nonzero. Then
exp(sv) · ξ =
X
esα(v) ξα .
α>0
s
Substitute e by t, we get
λ(t) · ξ =
X
tα(v) ξα .
α>0
Since α(v) > 0, ξ is unstable.
All other points are strictly semistable. For example, regular semi-simple
elements have closed orbits of maximal dimension but their stabilizers are
maximal tori of G.
Exercises
1. Let G be any reductive group with Lie algebra g. G acts on g by adjoint
action. What is the categorical quotient g//G? Is it geometric? (Note:
Exercise 2.3.2 is a special case of this problem.)
2. Let T be a torus of dimension r. Let L be a very ample linearized ample
line bundle defining an equivariant embedding X ⊂ P(V ) where V =
Γ(X, L)∗ . Then V splits into the direct sum of eigensubspaces
M
V =
Vχ .
χ∈χ(T )
42
For any x ∈ X, define
st(x) = {χ|x̂ =
X
vχ , vχ 6= 0}
χ
which is called the state of x, and
st(x) = convex hull of {χ|x̂ =
X
vχ , vχ 6= 0}
χ
which is called the state polytope of x. Prove the following.
(1) st(x) does not depend on the choice of the lifting x̂;
(2) x ∈ X ss (L) if and only if 0 ∈ st(x);
(3) x ∈ X s (L) if and only if 0 belongs to the interior of st(x).
3. Back to Example 2.6.11. Let Xn,m denote the space of hypersurfaces of
degree m in Pn acted upon by PGLn+1 . Let P be the Newton polytope of
degree m homogeneous polynomials in (n+1) variables. For any hypersurface H defined by F = 0, let St(H) be the state of F , i.e., the subset of integral points of P that appear in F with nontrivial coefficients. Then prove
that H is semistable (resp. stable) if and only if St(H) do not lie below
(resp. strictly below) any hyperplane through the center of the Newton
polytope.
4. Prove the following for the numerical function µL (x, λ).
(1) Prove that Definitions 3.2.1 and 3.2.2 are equivalent when the linearization L is very ample.
(2) For any x ∈ X and λ ∈ χ(G)∗ , set y = limt→0 λ(t)·x. Then µL (x, λ) =
µL (y, λ).
(3) µL (g · x, λ) = µ(x, g −1 λg) for any g ∈ G and λ ∈ χ(G)∗ .
(4) For any x ∈ X and λ ∈ χ(G)∗ , the map
PicG (X) → Z, L → µL (x, λ)
is homomorphism of groups.
(5) Let
P (λ) = {g ∈ G| lim λ(t)−1 gλ(t) exists in G}.
t→0
This is a parabolic subgroup of G, having
L(λ) = {ḡ = lim λ(t)−1 gλ(t)|g ∈ P (λ)}
t→0
as its Levi subgroup. In fact, let
U (λ) = {g ∈ G| lim λ(t)gλ(t) = 1.}
t→0
Then U (λ) is the unipotent radical of P (λ) and
P (λ) = L(λ) n U (λ).
Prove that µ(x, g −1 λg) = µL (x, λ) for ant g ∈ P (λ).
43
(6) If f : X → Y is a G-equivariant morphism, then for M ∈ PicG (Y ),
∗
µf M (x, λ) = µM (f (x), λ.
5. Let T be the set of all maximal tori of G and for any T ∈ T , let LT denote
the induced T -linearization. Show that
\
X ss (L) =
X ss (LT ),
T ∈T
X s (L) =
\
X s (LT ).
T ∈T
6. Let a reductive group G act on a scheme X. Then the G-action is proper
if and only if for every one-parameter subgroup λ, the induced λ-action is
proper. (Hint: use Iwahori’s theorem and the valuation criterion of properness.)
44
3.3. The Kempf-Ness analytic criterion.
In this subsection, we assume that the base field is C.
Again, consider a very ample linearization L and an induced equivariant embedding X ⊂ P(V ) where V = H 0 (X, L)∗ .
Let K ⊂ G be a compact form of G. Choose an Hermitian metric on V
invariant under K. For any point v ∈ V , Kempf and Ness consider the
following function
pv : G → R
pv : g → ||g · v||2 .
Lemma 3.3.1. (Kempf-Ness lemma.)
(1) if g is a critical point of pv , then it is a minimum;
(2) if pv attains a minimum, it does so on exactly one double coset KgGv ∈
K\G/Gv ;
(3) pv attains a minimum if and only if G · v is closed in V .
Proof. We will prove (1) and (2) altogether. This is divided into two steps.
First, we prove the case when G is a torus. Second, we use Cartan’s decomposition to reduce the general case to the toroidal case.
Step 1. Assume that G is a torus T ∼
= C∗ . Then we have
V = ⊕χ Vχ
where χ : G → C∗ is a character of T . Any character is necessarily of the
form
χ : G → k∗
Y
i
g = (t1 , . . . , tn ) →
tm
i
i
where (m1 , . . . , mn ) is a sequence of integers. For any v ∈ V , we can write
X
v=
vχ
according to the decomposition V = ⊕χ Vχ . Define
st(v) = {χ : vχ 6= 0},
called the state of v. Then
g·v =
X Y
χ∈st(v)
i
tm
i vχ
i
where χ is identified with (m1 , . . . , mn ). Hence
X
Y
pv (g) =
||vχ ||2
|ti |2mi .
i
χ∈st(v)
45
Since pv is K-invariant, pv induces a function
log
qv
G/K −−∼−→ Rn −−−→ R
=
given by
log : [(t1 , . . . , tn )] −→ (log|t1 |, . . . , log|tn |)
X
P
qv
(x1 , . . . , xn ) −→
eaχ +2 i mi xi
χ∈st(v)
where aχ = log||vχ ||2 . Note also that
Y
i
Gv = {(t1 , . . . , tn ) :
tm
i = 1, for all (mi ) ∈ st(v)}
i
whose image in G/K is
{(x1 , . . . , xn ) :
X
mi xi = 0, for all (mi ) ∈ st(v)}.
i
Now some very elementary computations on qv will easily verify both (1)
and (2).
Step 2. Now we return to the general case.
Let T be the set of all maximal algebraic tori T in G such that K ∩ T is
the maximal compact torus subgroup KT of T . Then by the Cartan decomposition (see, e.g., [31]), we have
G/K = ∪T ∈T T /KT .
As ph·v (g) = pv (gh), we may well assume that e is a critical point of pv .
Let g ∈ G, then g = kt where k ∈ K, t ∈ T for some T ∈ T . By the left
K-invariance of pv , we have pv (g) = pv (t). As e must be a critical point of
pv |T as well, we have
pv (g) = pv (t) ≥ pv (e)
by the toroidal case of (1). That is, e is a minimum.
To prove (2), we may assume again that pv attains its minimum at e. Let
m be the set of points where this minimum is attained. Then clearly we
have
KGv ⊂ m.
On the other hand, let g ∈ m, we can write g = kt as before. Then t is
a point where pv |T attains its minimum pv (e). Hence by the toroidal case,
t ∈ KT Tv . Hence
m ⊂ K(∪T ∈T KT Tv ) ⊂ KGv .
This proves (2).
We now start to prove (3). It suffices to prove
Claim: if G · v is not closed, then inf pv can not be attained.
46
So, assume that G · v is not closed. We can find a parabolic subgroup
P of G such that every maximal torus T of P contains a one-parameter
subgroup λ : C∗ → T such that limt→0 λ(t) · v exists in V and is a point
outside of G · v (see [7] or [56]).
Let P̄ be the parabolic subgroup of G which is conjugate to P under the
real structure on G defined by K. Then P ∩ P̄ contains a maximal torus S
defined over R and S is in the collection T .
Through the one-parameter subgroup λS of S, we have an action of C∗
on V such that λS · v is not closed and the maximal compact subgroup of
C∗ preserves the Hermitian form on V . As the claim for λS ∼
= C∗ implies
the claim for G, we assume now that G = λS .
As before, we can write
qv =
X
bi eli x
with bi > 0. As limx→−∞ qv (x) exists, we must have li ≥ 0. Since limt→0 λ(t)·
v 6= v, some li must be positive. Thus qv (x) is strictly increasing, hence it
(thus pv as well) never obtains a minimal value. This proves the claim. ¤
As a consequence, we obtain
Theorem 3.3.2. (Kempf-Ness criterion)
(1) x is semistable if and only if inf px̂ 6= 0;
(2) x is polystable if and only if px̂ attains its minimum on G if and only if px̂
has a critical point on G.
(3) x is stable if and only if px̂ attains its minimum and Gx̂ is finite if and
only if px̂ has a critical point on G and Gx̂ is finite.
Proof. (1) is the opposite of the statement “x is unstable if and only if
infpx̂ = 0” which is obvious because x is unstable if and only if 0 ∈ G · x̂
by part (3) of Theorem 3.1.1.
(2). Note that x is polystable if and only if G · x̂ is a closed orbit (see
Theorem 3.1.1). Hence (2) follows directly from Lemma 3.3.1.
(3) follows from the fact that x is stable if and only if x is polystable and
Gx̂ is finite.
¤
For G = SL(V ), one can restate the above theorem as follows, using the
fact that SL(V) acts transitively on the space Mω (V ) of Hermitian metrics
on V with a given volume form ω.
Corollary 3.3.3. Let V be a vector space over C with a volume form ω. Let
W ⊂ V ⊗n be a representation of SL(V ), x ∈ W a nonzero vector. Then the
following are equivalent:
(1) x is polystable, that is, the orbit SL(V ) · x ⊂ W is closed;
47
(2) There is a Hermitian metric m0 on V with volume form ω which minimizes the function
dx (m) = |x|2m
on Mω (V ) where |x|m is the length of x under the metric m;
(3) There is a metric m0 on V with a volume form ω which is a critical point
for the function dx : Mω (V ) → R.
Exercises
1. Let
qv : Rn −−−→ R
X
P
(x1 , . . . , xn ) −→
eaχ +2 i mi xi
χ∈st(v)
be as in the proof of Lemma 3.3.1. Show that
(1) if (x1 , . . . , xn ) is a critical point of qv , then it is a minimum;
(2) if qv attains a minimum at (x1 , . . . , xn ) and (y1 , . . . , yn ), then
X
mi (yi − xi ) = 0, for all χ = (mi ) ∈ st(v).
i
48
3.4. Moment map.
A smooth manifold M equipped with a non-degenerate closed 2-form
ω is called a symplectic manifold, and in this case, the 2-form ω is called
a symplectic form. Let K be a compact Lie group. We say that K acts
symplectically on M if for all k ∈ K, we have k ∗ (ω) = ω.
Given any ξ ∈ k = Lie K, it generates a vector field ξM ∈ T M defined by
d
(exp(tξ) · m)|t=0 .
dt
For simplicity, we also use ξ to denote the vector field ξM if confusion is
unlikely.
ξM,m =
Definition 3.4.1. A moment map for the action of K on M is a smooth map
Φ : M → k∗ such that it is K-equivariant with respect to the given action
on M and the co-adjoint action of K on k∗ , and satisfies
dΦξ = ιξ ω
for all ξ ∈ k, where Φξ = hΦ, ξi is the projection along ξ.
If a moment map exists then it is defined uniquely up to the addition of
a constant from k∗ fixed by the coadjoint action. In particular, it is unique if
K is semi-simple. A moment map exists if H 1 (X, R) = 0. In the projective
case, it always exists as we explain now.
In connection with GIT, consider the situation when ω is a Hodge form,
that is, there is an ample line bundle L over M such that c1 (L) = [ω]. By
taking a large tensor power Ln (or equivalently scaling the form ω by a
multiple n), we may assume that L is very ample and defines an embedding of M into a projective space P(V ) where V = Γ(X, L)∗ . For simplicity,
let us assume that ω is induced from the Fubini-Study metric of P(V ).
Putting things algebraically, let G be the complexification of a compact
Lie group K. Then G is necessarily a reductive algebraic group over the
field of complex numbers C. Assume that G acts on a projective nonsingular algebraic variety X ⊂ P(V ) via a linear representation on V . One can
choose a positive definite Hermitian inner product h , i on V such that K
acts on V by means of a homomorphism ρ : K → U (V ), where U (V ) is the
unitary group of V . Let ω be the real 2-form corresponding to a FubiniStudy metric on P(V ) determined by the inner product on V . We normalize it by requiring that, for any local holomorphic lifting z : U → V \ {0}
of an open subset U of P(V ),
ω = 2i∂ ∂¯ log ||z||2 .
The form ω is a symplectic (actually Kähler) form and K acts on (P(V ), ω)
symplectically.
49
There is a moment map P(V ) → u(V )∗ where u(V ) = Lie(U (V )) defined
by
hx̂, ξP(V ) x̂i
Φ(x)(ξ) =
2πi||x̂||2
for all ξ ∈ u(V ), where x̂ is the vector of projective coordinates of x and we
identify the vector field ξP(V ) with an element of the Lie algebra of U (V )
represented by a skew-Hermitian operator. It induces a canonical moment
map Φ : X → k∗ making the following diagram commutes
X −−−→ P(V )




y
y
k∗ ←−−− u(V )∗
Note that this map depends on the choice of the representation
G → GL(V )
which is the choice of a linear action on the very ample line bundle L.
Remark 3.4.2. More precisely, we have
(1) The so-defined moment maps are in one-to-one correspondence
with linearized very ample line bundles. When the underlying
line bundle is fixed, then choice of an additive constant for moment
map corresponds to a choice of a character used to twist the linear
action (see the Exercise 1 below).
(2) If L is not necessarily very ample but ample, we can define ΦL by
1 L⊗n
Φ , where L⊗n is very ample. Thus all the above can be exn
tended to moment maps defined by ample line bundles and rational character.
Exercises
1. Verify that
hx̂, ξP(V ) x̂i
2πi||x̂||2
satisfies the definition of moment map.
Φ(x)(ξ) =
2. Let L be a very ample linearized line bundle which induces a G-action
on V = Γ(X, L)∗ by a homomorphism
ρ : G −→ GL(V ).
∗
Let χ : G → C be a character of G and we twist the linearization L by χ
to obtain a new linearization L(χ). That is, under the linearization L(χ), G
acts on V = Γ(X, L)∗ given by
ρ̃ : G −→ GL(V )
50
ρ̃(g) = χ(g)ρ(g).
Verify that under suitable identification between characters of G and central elements of k∗ , we have
ΦL(χ) = ΦL + 2πiχ.
(Hint: dρ̃(ξ) · v = dρ(ξ) · v + 2πihχ, ξiv.)
51
3.5. The moment map criterion.
First, we need to observe from the definition that
Lemma 3.5.1. We have
dpx̂ (e)
(1) Φ(x) = 4πi||x̂||
2 . Consequently,
(2) dpx̂ (g) = 0 if and only if Φ(g · x) = 0.
Proof. (1). For any ξ ∈ Te G = g,
d
dpx̂ (e)(ξ) = hexp(tξ)x̂, exp(tξ)x̂i|t=0 = 2hx̂, ξ x̂i.
dt
Hence we obtain
dpx̂ (e)
Φ(x) =
.
4πi||x̂||2
(2). Because ph·x̂ = px̂ (gh), it suffices to prove that
dpx̂ (e) = 0 if and only if Φ(x) = 0.
But this follows from (1).
¤
Theorem 3.5.2. (The moment map criterion.)
(1) x is semistable if and only if 0 ∈ Φ(G · x);
(2) x is polystable if and only 0 ∈ Φ(G · x).
(3) x is stable if and only if 0 ∈ Φ(G · x) and Gx is finite.
Proof. (1). x is semistable if and only if G · x contains a polystable point y
if and only if, by Theorem 3.1.1 (2) and Theorem 3.3.2 (2), pŷ has a critical
point g if and only if, by Lemma 3.5.1, Φ(g·y) = 0 if and only if 0 ∈ Φ(G · x).
(2). x is polystable if and only if G · x̂ is closed if and only if px̂ attains its
minimum if and only if dpx̂ (g) = 0 for some g if and only if 0 ∈ Φ(G · x).
(3) follows immediately from (2).
¤
Exercises
1. Prove that the set X ps (L) of polystable points is G · Φ−1
L (0).
2. For a linearized ample line bundle L, it is called G-effective if X ss (L) 6=
∅. Show that L is G-effective if and only if the moment image ΦL (X) contains the origin. Also, give an example of an ample but not G-effective
linearized line bundle.
52
3.6. Relation with symplectic reduction.
ss
Theorem 3.6.1. Φ−1
L (0) is included in X (L) and the inclusion induces a homeomorphism
∼
=
Φ−1 (0)/K −−−→ X ss (L)//G.
Proof. By Theorem 3.5.2 (2), we have the inclusion and the inclusion clearly
induces a continuous map
γ : Φ−1 (0)/K −−−→ X ss (L)//G.
Surjectivity of this map follows from Theorem 3.5.2 (1) and (2), that is,
G · Φ−1
L (0) is precisely the set of polystable points.
To see the injectivity, let x, y ∈ Φ−1 (0) such that K · x 6= K · y. Since
G · x and G · y are closed orbits in X ss (L), it suffices to show that they are
disjoint. If not, there is g ∈ G such that y = g · x. By Lemma 3.5.1, e and g
are critical points of px̂ . Then by Lemma 3.3.1 (2), g ∈ KGx̂ . This implies
that K · x = K · y, a contradiction.
Thus γ is a continuous bijection from a compact space to a Hausdorff
space, hence is a homeomorphism.
¤
Example 3.6.2. Consider the action of C∗ on P3 by
λ · [x : y : z : w] = [λx : λy : λ−1 z : w].
The moment map is
Φ([x : y : z : w]) =
|x|2 + |y|2 − |z|2
.
|x|2 + |y|2 + |z|2 + |w|2
The image Φ(X) is [−1, 1] with three critical values −1, 0, 1. So, we consider the level sets Φ−1 (− 21 ), Φ−1 (0), Φ−1 ( 12 ). In Figure 5, we illustrate the
real parts Φ−1 (− 21 )R , Φ−1 (0)R , Φ−1 ( 12 )R of the level sets restricted to C3 ⊂ P3
(the C3 is defined by setting w = 1). It turns out this real picture preserves
all the topological information we need.
To understand this picture, note that the real points of S 1 are Z2 =
{−1, 1}. So, the real parts of
1
1
Φ−1 (− )/S 1 , Φ−1 (0)/S 1 , Φ−1 ( )/S 1
2
2
are
1
1
Φ−1 (− )R /Z2 , Φ−1 (0)R /Z2 , Φ−1 ( )R /Z2 .
2
2
53
P1
X −1/2
X0
X 1/2
Figure 5. Wall-Crossing Maps
Note that Z2 acts on each level set by identifying the lower part with the
upper part. Hence, the quotient can be naturally identified with the upper
part. Now observe that the left map happens to be an isomorphism, but
the right map is a (real) blowup along the origin so that the special fiber
is RP1 . Now complexifying this picture and compactifying the results, we
obtain three quotients: X[−1,0] ∼
= P2 , X{0} ∼
= P2 , and X[0,1] isomorphic to the
2
blowup of P along a point. The reader may try to classify the generic C∗ orbits and study their degenerations. He can verify that the Chow quotient
is also isomorphic to the blowup of P2 along a point.
Exercises
1. Let SL(2) act on the product of the projective lines, (P1 )n . Every linearized ample line bundle on (P1 )n is of the form ⊗ni=1 πi∗ OP1 (ri ) where π
is the projection from (P1 )n to the i-th factor and (r1 , · · · , rn ) is a sequence
of positive integers. Identify P1 with the unit sphere and su(2)∗ with R3 ,
compute the moment map for the SU (2) action on (P1 )n , and interpret
symplectic reductions and hence the GIT quotients as moduli spaces of
spatial polygons with fixed lengthes (r1 , · · · , rn ).
54
4. C ONFIGURATION OF LINEAR
SUBSPACES
The materials in this section are reproduced from the article [37].
Let V and W be two vector spaces. Consider the product of the Grassmannians
Πm
i=1 Gr(ki , V ⊗ W ).
The group SL(V ) acts diagonally on Πm
i=1 Gr(ki , V ⊗ W ) by operating on
the factor V . We will study the GIT of this action6.
4.1. Criteria for stable configuration.
To proceed we need a lemma.
Lemma 4.1.1. Let q be the vector (q1 , . . . , qn ) such that
(?) q1 ≥ q2 . . . ≥ qn , and q1 + . . . + qn = 0.
Let qs be the vector (q1 , . . . , qn ) such that
q1 = . . . = qs = n − s, qs+1 = . . . = qn = −s, for s = 1, . . . , (n − 1).
Then q is a linear combination of qs , s = 1, . . . , (n − 1), with nonnegative coefficients.
Proof. Indeed, one can check that
q1 − q 2
qn−1 − qn
q=
q1 + . . . +
qn−1 .
n
n
¤
Let ω = {ω1 , . . . , ωm } be a set of positive integers, and
∗
Lω = ⊗m
i=1 πi (OGr(ki ,V ⊗W ) (ωi ))
be the ample line bundle over Πm
i=1 Gr(ki , V ⊗ W ) associated with ω, where
πi is the projection from the product space to the i-th factor. This line
bundle has a unique SL(V )-linearization because SL(V ) is semisimple.
Theorem 4.1.2. A system of linear subspaces {Ki ⊂ V ⊗ W } as a point of
Πm
i=1 Gr(ki , V ⊗W ) is semistable (resp. stable) with respect to the SL(V )-linearized
invertible sheaf Lω if and only if, for all nonzero proper subspace H ⊂ V , we have
1 X
1 X
ωi dim(Ki ∩ (H ⊗ W )) ≤
ωi dim Ki
dim H i
dim V i
(resp. <).
6
The motivation to consider tensor product and, later, quotient linear spaces, is that
such a setup is directly related to the construction of moduli spaces of configurations of
coherent sheaves. Besides this, the problem itself seems to be of independent interest.
55
Proof. Choose a basis v1 , . . . , vn of V such that H = span{v1 , . . . , vs }. Set
Hi = span{v1 , . . . , vi }. In particular, we have Hn = V and Hs = H.
Let w1 , . . . , wd be a basis for W . We list the basis of V ⊗W made of vi ⊗wj
as
{v1 ⊗ w1 , v1 ⊗ w2 , . . . , vn ⊗ wd }.
Let Ei (1 ≤ i ≤ nd) be spanned by the first i vectors in the above basis.
Let K be any subspace of V ⊗ W . Then for any integer 1 ≤ j ≤ k =
dim K, there are integers lj such that
dim K ∩ Elj −1 = j − 1, dim K ∩ Elj = j.
Under the basis {v1 ⊗ w1 , v1 ⊗ w2 , . . . , vn ⊗ wd }, K can be represented by
a matrix


a11 · · · a1l1 0 · · ·
0 0···0
 a21 · · · · · · a2l2 · · ·
0 0···0 
 .

.
.
.
.
..
..
 ..

..
..
..
..
.
.
ak1 · · · · · · · · · · · · aklk 0 · · · 0
In the Plucker embedding, one sees that
pi1 ···ik (K) = 0 if ij > lj
pl1 ···lk (K) 6= 0.
(i)
We now apply the above to all Ki (1 ≤ i ≤ m) and let lj be the numbers
associated to Ki .
Next, consider the one-parameter subgroup λ(t) of SL(V ) defined by a
vector q = (q1 , . . . , qn ) as a diagonal matrix
λ(t) = diag(tq1 , . . . , tqn )
with
q1 + . . . + qn = 0.
By permutation if necessary, we can further assume that
q1 ≥ q2 . . . ≥ qn .
Let each qi repeat m times, we obtain a new diagonal matrix
0
0
λ0 (t) = diag(tq1 , . . . , tqmn ).
Under this convention and from the matrix representations of K, we see
that
0
0
pi1 ···ik (λ(t)K) = tqi1 +...qik pi1 ···ik (K).
Hence by the minimality of the numerical function we obtain
Lω
µ ({Ki }, λ) =
m
X
i=1
56
ωi
ki
X
j=1
ql0i .
j
(i)
Using the fact that dim Ki ∩ Ej − dim Ki ∩ Ej−1 equals 0 when j 6= lj
and equals 1 otherwise, we can rewrite
Lω
µ ({Ki }, λ) =
m
X
dn
X
ωi (
qj0 (dim Ki ∩ Ej − dim Ki ∩ Ej−1 )).
i=1
j=1
(Note here that µLω ({Ki }, λ) is linear in (q1 , . . . , qn ). This observation will
be useful later.)
Now replace λ by the one-parameter subgroup λs defined by qs (1 ≤
s ≤ (n − 1), see Lemma 4.1.1), then we have
Lω
µ ({Ki }, λs ) =
m
X
sd
X
ωi ( (n − s)(dim Ki ∩ Ej − dim Ki ∩ Ej−1 ))
i=1
−
m
X
i=1
ωi (
j=1
nd
X
s(dim Ki ∩ Ej − dim Ki ∩ Ej−1 )).
j=sd+1
After cancelation, we obtain
m
X
Lω
µ ({Ki }, λs ) =
ωi ((n−s) dim Ki ∩Esd −s(dim Ki ∩Edn −dim Ki ∩Esd )).
i=1
That is,
µLω ({Ki }, λs ) =
m
X
ωi (n dim Ki ∩ Esd − s dim Ki ∩ Edn ),
i=1
Noting that Esd = H ⊗ W and End = V ⊗ W , we have
m
m
X
X
Lω
µ ({Ki }, λs ) = dim V
ωi dim Ki ∩ (H ⊗ W ) − dim H
ωi dim Ki .
i=1
i=1
Now if {Ki } is Lω -semisimple (resp. simple), then
µLω ({Ki }, λs ) ≤ 0
(resp. <) which is the same as that
1 X
1 X
ωi dim(Ki ∩ (H ⊗ W )) ≤
ωi dim Ki
dim H i
dim V i
(resp. <).
Conversely, if the inequality
1 X
1 X
ωi dim(Ki ∩ (H ⊗ W )) ≤
ωi dim Ki
dim H i
dim V i
holds for all H, but {Ki } is not Lω -semistable. Then there is one-parameter
subgroup λ such that µLω ({Ki }, λ) > 0. By conjugation and permutation,
we can assume that the vector q that defines λ satisfies (?) (see Lemma
57
4.1.1). Note that such a vector q is a linear combination of qs (1 ≤ s ≤ n−1)
with non-negative coefficients. Note also from the above that µLω ({Ki }, λ)
is linear in q. Hence there must exists s (1 ≤ s ≤ n−1) such that µLω ({Ki }, λs ) >
0, but this is equivalent to that
1 X
1 X
ωi dim(Ki ∩ (H ⊗ W )) >
ωi dim Ki
dim H i
dim V i
for some vector subspace H, a contradiction.
Similarly, if the strict inequality
1 X
1 X
ωi dim(Ki ∩ (H ⊗ W )) <
ωi dim Ki
dim H i
dim V i
holds for all H, then by the above {Ki } is Lω -semistable. Assume that it
is not stable. Then there is a λ that satisfies (?) of Lemma 4.1.1 such that
µLω ({Ki }, λ) = 0. Then the same arguments as above plus that we already
know µLω ({Ki }, λs ) ≤ 0 will yield that there is s (1 ≤ s ≤ n − 1) such that
µLω ({Ki }, λs ) = 0, but this is equivalent to that
1 X
1 X
ωi dim(Ki ∩ (H ⊗ W )) =
ωi dim Ki
dim H i
dim V i
for some vector subspace H, a contradiction.
This completes the proof.
¤
The above theorem can also be equivalently stated in terms of quotients.
We will use the notation Gr(V ⊗ W, a) for the Grassmannian of quotient
linear spaces of V ⊗ W of dimension a.
Let ω be a set of positive integers and
∗
L0ω = ⊗m
i=1 πi (OGr(V ⊗W,ai ) (ωi ))
be the ample line bundle over Πm
i=1 Gr(V ⊗ W, ai ) defined by ω where πi is
the projection from the product space to the i-th factor.
fi
Theorem 4.1.2’. A configuration {V ⊗ W → Ui → 0} as a point of Πm
i=1 Gr(V ⊗
W, ai ) is semistable (resp. stable) with respect to the SL(V )-linearized invertible
sheaf L0ω if and only if, for all nonzero proper subspace H ⊂ V , we have
m
m
1 X
1 X
ωi dim Ui ≤
ωi dim fi (H ⊗ W )
dim V i=1
dim H i=1
(resp. <). In particular, fi (H ⊗ W ) 6= 0 for some i.
We note that when m = 1, this is Simpson’s Proposition 1.14 of [82],
where it is used to construct the moduli space of coherent sheaves.
For the action of SL(V ) on Gr(V ⊗ W, a), if there is a GIT quotient,
then it will be unique because there is only one ample line bundle over
58
Gr(V ⊗ W, a) up to homothety, and, this line bundle has a unique SL(V )
linearization. There could be none, for example, this will be the case when
dim W = 1. From now on, we assume that a GIT quotient exists and we
use M to denote this unique quotient variety.
Fix a set of positive numbers ω = {ω1 , . . . , ωm } and let Mω be the quotient variety of the locus of the L0ω -semistable configurations.
Proposition 4.1.3. Fix an integer 1 ≤ i ≤ m. For sufficiently large ωi (relative
to other ωj ), we have
(1) If a configuration {V ⊗ W → Uj → 0} is L0ω -semistable, then its i-th
component V ⊗ W → Ui → 0 is also semistable.
(2) If the i-th component of a configuration {V ⊗ W → Uj → 0} is stable,
then the configuration {V ⊗ W → Uj → 0} is L0ω -stable.
(3) In particular, there is a surjective projective morphism from Mω to M
πi : Mω → M.
Similar statements hold in terms of systems of subspaces.
Proof. For any subspace H ⊂ V , the inequality
1 X
1 X
ωj dim Uj ≤
ωj dim fj (H ⊗ W ) (resp <)
dim V j
dim H j
holds if and only if
dim H dim Ui − dim V dim fi (H ⊗ W ) ≤
X
X
1
ωj dim fj (H ⊗ W )) (resp <)
ωj dim Uj − dim V
(dim H
ωi
j6=i
j6=i
holds.
Let R be the right hand term of the last inequality. Then we can choose
a sufficiently large ωi , such that |R| < 1. Now if the inequality “≤” holds,
the left hand term
dim H dim Ui − dim V dim fi (H ⊗ W )
must be nonpositive since it is an integer. This proves (1).
For (2), if dim H dim Ui − dim V dim fi (H ⊗ W ) < 0, then
dim H dim Ui − dim V dim fi (H ⊗ W ) ≤ −1.
Since we have |R| < 1, we obtain
dim H dim Ui − dim V dim fi (H ⊗ W ) < R
which implies
1 X
1 X
ωj dim Uj <
ωj dim fj (H ⊗ W ).
dim V j
dim H j
59
(3). The existence of the morphism πi : Mω → M follows from (1). The
surjectivity follows from (2).
¤
Exercises
1. Consider the action of SL(2) on X = (P1 )n . Every linearized ample line
bundle on (P1 )n is of the form Lr = ⊗ni=1 πi∗ OP1 (ri ) where π is the projection
from (P1 )n to the i-th factor and r = (r1 , · · · , rn ) is a sequence of positive
integers. Show that
(1) (x1 , · · · , xn ) ∈ (P1 )n is Lr is semistable (resp. stable) if and only if
for every point x ∈ (P1 )n , or equivalently, for every x ∈ {x1 , · · · , xn },
we have
X
1X
ri ≤
ri . (resp. <)
2 i
x =x
i
In particular, when all ri are equal, this means (x1 , · · · , xn ) is semistable
(resp. stable) if and only if at most (resp. no more than) half of the
points coincide.
P
(2) X ss (Lr ) 6= ∅ if and only if 2ri ≤ j rj for all i.
2. Let r = (r1 , · · · , rn ) be a sequence of positive real numbers. Prove that
(r1 , · · · ,P
rn ) can be realized as the side lengths of an n-gon in R3 if and only
if 2ri ≤ j rj for all i.
60
4.2. Harder-Narasimhan and Jordan-Hölder filtrations.
Theorem 4.1.2 motivates the following definition. Let ω = (ω1 , . . . , ωm )
be a set of positive numbers,
Q called the weights. The normalized total weighted
dimension of K = {Ki } ∈ i Gr(ki , V ⊗ W ) with respect to ω is defined by
1 X
℘ω (K) =
ωi dim Ki .
dim V i
For any subspace H of V , there is an induced subconfiguration of linear
subspaces in H ⊗ W
H = (K1 ∩ (H ⊗ W ), . . . , Km ∩ (H ⊗ W ))
whose normalized total weighted dimension with respect to ω is
1 X
℘ω (H) =
ωi dim Ki ∩ (H ⊗ W ).
dim H i
Definition 4.2.1. The configuration K is ℘ω -semistable (resp. stable) with
respect to the weights ω if
℘ω (H) ≤ ℘ω (K) (resp. <)
for every subspace H of V .
Then, Theorem 4.1.2 can be restated as
Theorem 4.2.2. The configuration K is GIT semistable (stable) with respect to
the linearized line bundle Lω if and only if it is ℘ω -semistable (stable) with respected to the weight set ω.
Let f : V → Q be a linear map. Then, the induced map V ⊗ W → Q ⊗ W ,
still denoted by f , induces a configuration {f (Ki )} of linear subspaces in
Q ⊗ W . A subconfiguration of {Ki } is the one induced from an inclusion
map i : H ,→ V .
Lemma 4.2.3. Let {Ki } be a configuration of linear subspaces of V ⊗ W and
0→F →V →Q→0
be an exact sequence. Let F and Q be the inducing configurations. Then
(1) ℘ω (Q) ≥ ℘ω (V)(resp. >) if ℘ω (F) ≤ ℘ω (V)(resp. >);
(2) ℘ω (F) ≥ ℘ω (Q)(resp. >) if ℘ω (F) ≥ ℘ω (V)(resp. >).
Proof. We prove (2) and leave (1) for the reader.
Let Fi be Ki ∩ (F ⊗ W ) and Qi be the image of Ki under the map f :
V ⊗ W → Q ⊗ W for all i. If ℘ω (F) ≥ ℘ω (V), then
1 X
ωi dim Fi ≥
dim F i
61
X
X
1 X
1
ωi dim Ki =
(
ωi dim Fi +
ωi dim Qi ).
dim V i
dim V i
i
Hence
X
X
dim F + dim Q X
ωi dim Fi ≥ (
ωi dim Fi +
ωi dim Qi ).
dim F
i
i
i
Therefore
X
dim Q X
ωi dim Fi ≥
ωi dim Qi .
dim F i
i
That is, ℘ω (F) ≥ ℘ω (Q).
The strict inequality can be proved similarly.
¤
Definition 4.2.4. For any configuration {Ki } of linear subspaces of V ⊗ W ,
if there is a filtration
0 = V 0 ⊂ V 1 ⊂ ··· ⊂ V h = V
with the inducing subconfigurations
(0)
(h)
(1)
{0} = {Ki } ⊂ {Ki } ⊂ · · · ⊂ {Ki } = {Ki }, 1 ≤ i ≤ m
(l)
where Ki = Ki ∩ (V l ⊗ W ) (1 ≤ l ≤ h) such that the quotient configura(l−1)
(l)
}i (1 ≤ l ≤ h) is ℘ω -semistable and the normalized total
tion {Ki /Ki
weighted dimension
X
1
(l−1)
(l)
), 1 ≤ l ≤ h
ωi dim(Ki /Ki
dim V l /V l−1 i
is strictly decreasing, then the filtration
0 = V 0 ⊂ V 1 ⊂ ··· ⊂ V k = V
or rather the filtered configuration
(0)
(1)
(h)
{0} = {Ki } ⊂ {Ki } ⊂ · · · ⊂ {Ki } = {Ki }, 1 ≤ i ≤ m
will be called a Harder-Narasimhan filtration for {Ki }.
Proposition 4.2.5. For every configuration {Ki } of linear subspaces of V ⊗ W ,
the Harder-Narasimhan filtration exists and is unique.
Proof. Let H be a subspace of V such that
1 X
℘ω (H) =
ωi dim Ki ∩ (H ⊗ W )
dim H i
is the maximal. If H = V , then K is ℘ω -semistable, we are done. Otherwise,
by maximality, H is ℘ω -semistable. Now assume H1 is another linear subspace such that ℘ω (H1 ) is maximal, that is, ℘ω (H1 ) = ℘ω (H). Then H ⊕ H1
is ℘ω -semistable of ℘ω (H ⊕ H1 ) = ℘ω (H). Consider the addition map
f : H ⊕ H1 → V.
62
Since H ⊕ H1 is ℘ω -semistable, we have that the normalized weighted dimension of the kernel Ker(f ) is less than or equal to ℘ω (H), therefore the
normalized weighted dimension of the image H + H1 is greater than or
equal to ℘ω (H) (by Lemma 4.2.3 (1)) and hence equal to ℘ω (H) by the maximality of ℘ω (H). This showed that there is a unique subspace V 1 such
that ℘ω (K1 ) is largest, where K1 is the induced configuration from V 1 . This
constitutes the first step of the filtration
0 ⊂ V 1 ⊂ V.
Next consider V /V 1 . If K/cK 1 is semistable, we are done because Lemma
4.2.3 (2) implies that ℘ω (K1 ) > ℘ω (K/K1 ). If K/K1 is not semistable, the
above procedure can be applied word for word to produce a unique linear
subspace V 2 (V 1 ⊂ V 2 ⊂ V ) with K2 /K1 semistable. By Lemma 4.2.3 (2)
again, ℘ω (K1 ) > ℘ω (K2 /K1 ) because ℘ω (K1 ) > ℘ω (K2 ). Hence by induction, we will obtain the desired filtration.
The uniqueness is clear from the proof.
¤
Definition 4.2.6. Assume that {Ki } is ℘ω -semistable. If there is a filtration
0 = V 0 ⊂ V 1 ⊂ ··· ⊂ V k = V
with the inducing subconfigurations
(0)
(h)
(1)
{0} = {Ki } ⊂ {Ki } ⊂ · · · ⊂ {Ki } = {Ki }, 1 ≤ i ≤ m
(l)
(l−1)
}i are ℘ω -stable with the same
such that the quotient systems {Ki /Ki
normalized total weighted dimension ℘ω (V), then the filtration is called a
Jordan-Hölder filtration.
Proposition 4.2.7. For any ℘ω -semistable {Ki }, a Jordan-Hölder filtration exists.
Proof. A construction goes as follows. If K is stable, we are done. Otherwise, let H be a maximal subspace such that ℘ω (H) = ℘ω (K). Then H must
also be semistable. Applying Lemma 4.2.3 (1) and (2), one can check that
K/H is ℘ω -stable and ℘ω (K/H) = ℘ω (K). Repeat the same procedure to H,
we will obtain a desired filtration.
¤
From the proof one see that a Jordan-Hölder filtration always exists but
depends on a choice of maximal subspaces H, hence it needs not to be
unique.
Exercises
1. (Splitting and Merging.) Let
K = (K1 , . . . , Km ) ∈ Gr(k1 , V ⊗ W ) × . . . × Gr(km , V ⊗ W )
63
be a configuration of vector subspaces of V = Cn with weightes ω =
(ω1 , . . . , ωm ). If for every i, ωi = si + ti where si and ti are nonnegative
integers, then we can split Ki with weight ωi into Ki with weight si and
e with new
Ki with weight ti . In this way, we obtain a new configuration K
weights ω̃. We may call such a process splitting or separation. Conversely,
as opposed to splitting, one may consider “merging”. That is, for any
e with weights ω
configuration of vector subspaces K
e , if K̃i = K̃j for some
i 6= j, then we can merge the two as one and count it with new weight
ωi = ω̃i + ω̃j . This way, we obtain a new configuration K with new weights
ω. We may call such a process merging. Prove that
(1) In either splitting or merging, we have that
e
℘ω (K) = ℘ω̃ (K),
and for any subspace H ⊂ K,
e
℘ω (H) = ℘ω̃ (H).
(2) In particular, K is semistable (stable) with respect to ω if and only if
e is semistable (stable) with respect to ω̃.
K
Qm
(3) Let Mω denote the GIT quotient of X =
i=1 Gr(ki , V ⊗ W ) by
SL(V ), defined by the SL(V )-linearized line bundle Lω . Then there
is natural closed embedding from the GIT quotient Mω to the corresponding GIT quotient Mω̃ .
(4) For any weights ω, if we write every ωi as the sum 1 + · · · + 1 (ωi
many), then we obtain a new weight set I = (1, . . . , 1) which we
shall call the trivial weight. Then every Mω can be embedded in MI
as a closed subvariety.
64
4.3. Polystable configurations.
In this section, we will focus on the special case when dim W = 1. So, let
V = (V1 , . . . , Vm ) ∈ Gr(k1 , V ) × . . . × Gr(km , V )
be a configuration of vector subspaces of V = Cn and ω = (ω1 , . . . , ωm ) be
a set of positive numbers.
Proposition 4.3.1. If V is ℘ω -semistable and is a direct sum ⊕li=1 Hi of a finite
number of subconfigurations, then all Hi and V have the same normalized total
weighted dimension. In particular, all Hi are also semistable.
Proof. We first prove the case when V = H1 ⊕ H2 . We have
0 → H1 → V → H2 → 0
and
0 → H2 → V → H1 → 0.
Since V is semistable, ℘ω (H1 ) ≤ ℘ω (V) and ℘ω (H2 ) ≤ ℘ω (V). By Lemma
4.2.3, ℘ω (H2 ) ≥ ℘ω (V) and ℘ω (H1 ) ≥ ℘ω (V). Hence they are all equal.
In general, write V = Hi ⊕ (the rest), by the case l = 2, ℘ω (Hi ) = ℘ω (V)
for every i.
¤
Definition 4.3.2. A semistable configuration V = (V1 , . . . , Vm ) is called
polystable if it is a direct sum of a finite number of stable subconfigurations of the same normalized total weighted dimension.
Proposition 4.3.3. V is polystable if and only if as a point in the product of the
Grassmannians its orbit is closed in the semistable locus.
Proof. Suppose that V = {Vi } is polystable and is the direct sum of stable
subconfigurations {Hq } induced from the decomposition V = ⊕q Hq . Let
V(t) be a curve in G · V for t near t0 . Let V(0) be the limit of V(t) in the
semistable locus at t0 . Then V(0) is the direct sum of {Hq (0)} where Hq (0)
is in the closure of G·Hq . By Proposition 4.3.1, Hq (0) is semistable. Since Hq
is stable, Hq (0) in the orbit G·Hq . This means there is a linear isomorphism
lq of V sending Hq to Hq (0) and inducing isomorphisms between Hq ∩ Vi
and Hq (0) ∩ Vi for all i. Since V is the direct sum ⊕q Hq , one can build a
linear isomorphism l of V from lq |Hq (for all q), sending Hq to Hq (0) for
all q and inducing isomorphisms between Hq ∩ Vi and Hq (0) ∩ Vi for all i.
Hence V(0) = {Hq (0)}q is in the orbit G · V. This shows that the orbit G · V
is closed in the semistable locus.
Conversely if V is a semistable configuration and the orbit G · V is closed
in the semistable locus, we need to show that V is polystable. Let F ⊂ V be
a subspace such that {F ∩ Vi } constitute the first step in the Jordan-Hölder
65
filtration of V. Choose a basis for F and extend it to a basis for V . Then
under this basis we can represent each Vi as an (n × ki ) matrix
µ
¶
Ai Bi
0 Di
where Ai generates F ∩ Vi . Let d be the dimension of F and λ(t) be a
one-parameter subgroup of GL(V ) defined by
µ
¶
tId
0
λ(t) =
0 In−d
Then we have
µ
¶
tAi tBi
λ(t)Vi =
0 Di
As t tends to zero, this splits the limit as the direct sum
F ∩ Vi ⊕ Q ∩ Vi
where Q is spanned by the basis element of V that are not in F . By Proposition 4.3.1, the configuration {F ∩ Vi ⊕ Q ∩ Vi }i is semistable. Since G · V
is closed in the semistable locus, this shows that {Vi } and
{F ∩ Vi ⊕ Q ∩ Vi }
are in the same orbit. Repeat the Jordan-Hölder process, this will eventually show that V is polystable.
¤
66
4.4. Balanced metrics and polystable configurations.
Definition 4.4.1. A Hermitian metric h on V is said to be a balance metric
for the weighted configuration (V, ω) of vector subspaces if the weighted
sum of the orthogonal projections from V onto Vi (1 ≤ i ≤ m) is the scalar
operator ℘ω (V) IdV . That is
m
X
ωi πVi = ℘ω (V) IdV
i=1
where πVi : V → Vi ,→ V is the orthogonal projection from V to Vi and IdV
is the identity map from V to V . In this case, we also say that the weighted
configuration (V, ω) is balanced with respect to the metric h. We say (V, ω)
can be (uniquely) balanced if there is a (unique) g ∈ SU(V )\ SL(V ) such
that (g · V, ω) is balanced.
Theorem 4.4.2. A configuration V = (V1 , . . . , Vm ) is polystable with respect to
a weight set ω if and only if there is a balance metric on V for the configuration.
Proof. First, it is easy to check that under the linearized line bundle Lω , the
moment map
√
Φ : Gr(k1 , V ) × . . . × Gr(km , V ) → −1 su(V )
of the diagonal action of SU(V ) is given by
X
Φ(V) =
ωi Ai A∗i − ℘ω (V)In
i
where V = (V1 , . . . , Vm ) ∈ Gr(k1 , V ) × . . . × Gr(km , V ) and Ai is a matrix
representation of Vi such that its columns form an orthonormal basis for
Vi (1 ≤ i ≤ m). (Here using an orthonormal basis {e1 , · · · , en } of V , we
∗
identify
√ su(V ) with su(n). Also, using the Killing form, we identify su(n)
with −1 su(n).)
Assume that V = (V1 , . . . , Vm ) is polystable with respect to the weighted
ω. By Proposition 4.3.3, its orbit in the semistable locus is closed. Hence
by, for example, Theorem 2.2.1 (1) of [17], there is an element g ∈ SL(V )
such that Φ(g · V) = 0. If g is the identity, this means that
X
ωi Ai A∗i = ℘ω (V)In
i
which is equivalent to
m
X
ωi πVi = ℘ω (V) IdV
i=1
because by a direct computation in linear algebra one can verify that the
orthogonal projection πVi can be identified with the matrix Ai A∗i under the
identification between V and Cn (using the orthonormal basis {e1 , · · · , en }).
67
That is, the standard hermitian metric h is a balance metric on V for the
configuration. Similarly, when g is not the identity, the hermitian metric
gh(•, •) = h(g•, g•) is a balance metric on V for the configuration.
Conversely, if there is hermitian metric h0 such that it is a balance metric
for the configuration V = (V1 , . . . , Vm ), then by scaling, we may assume
that h0 and h have the same volume form. Hence there is g ∈ SL(V ) such
that h0 = gh. This implies that
Φ(g · V) = 0.
Hence (again by, for example, Theorem 2.2.1 (1) of [17]), the orbit through
g · V is closed in the semistable locus. Therefore by Proposition 4.3.3, V is
polystable with respect to Lω .
¤
This theorem was previously known for the so-called m-filtrations with
the trivial weights I = (1, . . . , 1) and was proved by Klyachko ([60]) and
Totaro ([86]).
68
4.5. Stability of tensor product.
Of special interest is the so-called m-filtration. A filtration V • is a weakly
decreasing configuration of subspaces
V = V 0 ⊃ V 1 ⊃ . . . ⊃ {0}.
By a m-filtration, we mean a collection V • (s) of filtrations of V , for 1 ≤ s ≤
m.
In [24] (cf. also [86]), Faltings and Wüstholz defined a stability for mfiltration. Their definition coincides with our definition when considering
the m-filtration as a configuration of vector subspaces with trivial weights
I = (1, . . . , 1).
Conversely, if we treat each Vi as a (trivial) filtration V ⊃ V i ⊃ {0}
and use splitting process, then any configuration of vector subspaces with
weights ω can also be considered as a m-filtration with trivial weights I,
and again the two stabilities coincides.
Given two m-filtrations V • (s) and W • (s), 1 ≤ s ≤ m. We define the
tensor product (V ⊗ W )• (s) as
X
(V ⊗ W )l (s) =
V p (s) ⊗ W q (s).
p+q=l
•
•
If V (s) and W (s) have attached weights ω and ω 0 , then we will use the
splitting and merging method to give {(V ⊗ W )l (s)} the induced weights
ω̃.
Proposition 4.5.1. If V • (s) and W • (s) are ℘ω -semistable and ℘ω0 -semistable,
respectively, then (V ⊗ W )• (s) is ℘ω̃ -semistable.
Proof. This proposition marginally generalizes Theorem 1 of [86]. It also
follows from the proof of [86] using the splitting and merging method to
relate weighted filtrations with unweighted (or trivially weighted) filtrations.
¤
A different way to prove this may be done via calculating the moment
map of Gr(pq, V ⊗ W ) using the moment map of Gr(p, V ) and Gr(q, W ).
69
4.6. Generalized Gelfand-MacPherson correspondence.
4.6.1. Correspondence between orbits.
Choosing a basis of V , we can identify V with Cn . Then a k-dimensional
vector subspace E ⊂ V ∼
= Cn can be represented by a full rank matrix M
of size n × k. The group G = SL(n, C) acts on M from the left. The group
Gk = SL(k, C) acts on M from the right. Two such matrices represent the
same vector subspace if and only if they are in the same orbit of Gk . Let
0
Un,k
be the space of all full rank matrices of size n × k. Then Gr(k, Cn ) is
0
the orbit space Un,k
/Gk .
Now assume that n < k1 + . . . + km . Given a configuration of vector
subspaces
(V1 , . . . , Vm ) ∈ Gr(k1 , n) × . . . × Gr(km , n),
let (M1 , . . . , Mm ) be their corresponding (representative) matrices. Now,
think of
M = (M1 , . . . , Mm )
0
as a matrix of size n × (k1 + . . . + km ) and let Un,(k
be the space of
1 ,...,km )
matrices of size n × (k1 + . . . + km ) such that M and each of its block matrix
Mj is of full rank for 1 ≤ j ≤ m.
0
: one is the action of the group
There are two group actions on Un,(k
1 ,...,km )
G = SL(n, C) from the left; the other is the action of the product group
Gk1 ,...,km = S(GL(k1 , C) × . . . × GL(km , C)) ⊂ SL(k1 + · · · + km , C)
with each factor acting on the corresponding block from the right. For
0
simplicity, we sometimes use G0 to denote Gk1 ,...,km . Quotienting Un,(k
1 ,...,km )
by the group Gk1 ,...,km , we obtain
X = Gr(k1 , n) × . . . × Gr(km , n)
with the residual group G = SL(n, C) acting diagonally as usual; Quoti0
enting Un,(k
by the group G = SL(n, C), we obtain
1 ,...,km )
Y = Gr(n, k1 + . . . + km )
with the residual group G0 = Gk1 ,...,km acting block-wise.
It follows that
Proposition 4.6.1. There is a bijection between G-orbits on X and G0 -orbits on
Y . Indeed, there is a homeomorphism between the (non-Hausdorff) orbit spaces
X/G and Y /G0 .
When k1 = k2 = . . . = km = 1, X is (Pn−1 )m and G1,...,1 is a maximal
torus of SL(m, C). In this case, the proposition is the Gelfand-MacPherson
correspondence ([23]).
70
Proof. The correspondence exists because each set of orbits are in one-to0
one correspondence with G × G0 -orbits on Un,(k
.
¤
1 ,...,km )
4.6.2. Quotients in stages.
From the previous section one naturally expects that the correspondence
between orbits will induce a correspondence between the set of GIT quotients of
X = Gr(k1 , n) × . . . × Gr(km , n)
by the group G = SL(n, C) and the set of GIT quotients of
Y = Gr(n, k1 + . . . + km )
by the group G0 = Gk1 ,...,km . The detail of this goes as follows.
First, recall that any G-linearized ample line bundle on X = Gr(k1 , n) ×
. . . × Gr(km , n) must be of the form Lω for some weights ω. For the Grassmannian Y = Gr(n, k1 + . . . + km ), there is only one line bundle L = OY (1)
up to homothety. But the character group of GL(k1 , C) × · · · × GL(km , C)
can be identified with Zm . That is, each set ω of positive integers defines a
character
χω : GL(k1 ) × · · · × GL(km ) → C∗ .
Let L(χω ) be the ample line bundle OY (1) twisted by the character χω .
L(χω ) is linearized for GL(k1 , C) × · · · × GL(km , C) and hence for its subgroup G0 = S(GL(k1 , C) × · · · × GL(km , C)).
Theorem 4.6.2. There is a one-to-one correspondence between the set of GIT
quotients of X = Gr(k1 , n) × . . . × Gr(km , n) by the group G = SL(n, C) and the
set of GIT quotients of Y = Gr(n, k1 +. . .+km ) by the group G0 = Gk1 ,...,km . More
precisely, for any sequence ω of positive integers, we have a natural isomorphism
between X ss (Lω )//G and Y ss (L(χω ))//G0
When k1 = . . . km = 1, the theorem was previously proved by Kapranov using the standard Gelfand-MacPherson correspondence. Here we
reproduce his proof in the general case.
Proof. First, recall that the coordinate ring of Gr(k, Cn ) in the Plüker embedding can be identified with the ring of polynomials f in matrices M of
size n × k such that f (M · g) = f (M ) for all g ∈ SL(k, C). In particular, we
have that the section space Γ(Gr(k, Cn ), OGr(k,Cn ) (d)) can be identified with
{f (M )|f (tM ) = td f (M ), f (M · g) = f (M ), g ∈ SL(k, C)}
for all integers d > 0.
Now using the group GL(n, C) in place of SL(k, C), the above has an
equivalent but more concise expression as follows. Recall that the character group of GL(k, C) can be naturally identified with the group of integers
71
Z. For any integer d > 0, let
χd : GL(k, C) → C∗
be the corresponding character of GL(k, C). Then we have
Γ(Gr(k, Cn ), OGr(k,Cn ) (d)) = {f |f (M · g) = χd (g)f (M ), g ∈ GL(k, C)}.
This is because the two identities:
f (tM ) = td f (M ) and f (M · g) = f (M ), g ∈ SL(k, C)
can be combined together in the single identity
f (M · g) = χd (g)f (M ), g ∈ GL(k, C).
From the above and considering the ring of polynomials in matrices M
of size n × (k1 + . . . + km ) one checks that
Γ(X, Ldω ) = {f (M )|f (M · g) = χdω (g)f (M ), g ∈ GL(k1 ) × · · · × GL(km )}
and
Γ(Y, Ld (χω )) = {f (M )|f (g 0 · M ) = χd (g 0 )f (M ), g 0 ∈ GL(n, C)}.
Therefore by taking the projective spectrum of the invariants of
A = ⊕d Γ(X, Ldω )
under the action of the group GL(n, C) and by taking the projective spectrum of the invariants of
B = ⊕d Γ(Y, Ld (χω ))
under the action of the group
GL(k1 ) × · · · × GL(km ),
we see that the both quotients
X ss (Lω )//G and Y ss (L(χω ))//G0
can be naturally identified with the projective spectrum of the ring
R = ⊕d Rd
where
Rd = {f (M )|f (M · g) = χdω (g)f (M ), f (g 0 · M ) = f (M )}
for all g ∈ GL(k1 )×· · ·×GL(km ), g 0 ∈ SL(n, C). (Note that here we take g 0 ∈
SL(n, C) instead of g 0 ∈ GL(n, C). This is because the effect of the central
part of GL(n, C) is already reflected by the scalar matrices in GL(k1 )×· · ·×
GL(km ).)
This has established the desired correspondence.
72
¤
4.7. The Cone of effective linearizations.
4.7.1. Point Configurations on Pn−1 . Consider the diagonal action of SL(n)
on (Pn−1 )m = P(Cn ). Every linearized ample line bundle on (Pn−1 )m is of
the form Lr = ⊗ni=1 πi∗ OPn−1 (ri ) where π is the projection from (Pn−1 )m to
the i-th factor and r = (r1 , · · · , rn ) is a sequence of positive integers.
P
Theorem 4.7.1. X ss (Lr ) 6= ∅ if and only if ri ≤ n1 j rj for all i.
Proof. Let {Vj } ∈ X ss (Lr ) 6= ∅. Then for H = Vi , we have
1 X
1X
rj ≤
rj dim Vj ∩ H ≤
rj .
dim H j
n j
Conversely, let {Vj } ∈ P(Cn ) be in general linear position. For any subspace H ⊂ Cn , since dim Vj ∩ H is either zero or 1, and there are at most
dim H many of them are non-zero, we have
1 X
1X
rj dim Vj ∩ H ≤ max{rj } ≤
rj .
dim H j
n j
¤
As the stability depends on ω, so does the moduli. In this section, we
study the G-ample cone to pave a way for the study of the variation of
the moduli. In particular, we will introduce a family of new polytopes:
diagonal hypersimpleces.
Given a linearized line bundle L over X, it is called G-effective if X ss (L) 6=
∅. Not all of Lω are G-effective. The following should characterize the effective ample ones.
We will always assume that the group G = SL(V ) acts freely on generic
configuration of linear subspaces, that is, G acts freely on an open subset of
generic points
in Πm
i=1 Gr(ki , V ). This should be true when n < k1 + · · · + km
P
2
and n ≤ i ki (n − ki ).
Conjecture 4.7.2. Under the above (and perhaps some additional natural) conditions, we have
P
(1) X ss (Lω ) 6= ∅ if and only if ωi ≤ n1 i ki ωi for all 1 ≤ i ≤ m if and only
P
if max{ωi }i ≤ n1 i ki ωi ;
P
(2) X s (Lω ) 6= ∅ if and only if ωi < n1 i ki ωi for all 1 ≤ i ≤ m if and only if
P
max{ωi }i < n1 i ki ωi .
The necessary parts of both (1) and (2) are true.
73
Proof. (1). The necessary direction is easy. Assume that X ss (Lω ) 6= ∅ and
let V = {Vi } ∈ X ss (Lω ). We have that for all W ⊂ V ,
1 X
1X
ωj dim(Vj ∩ W ) ≤
ki ωi .
dim W j
n i
Now for any given i, take W = Vi , then we obtain
1 X
1X
ωi ≤
ωj dim(Vj ∩ W ) ≤
ki ωi
dim W j
n i
for all i.
The necessary part of (2) can be proved similarly.
¤
Equivalently,
P
Conjecture 4.7.3.
(1) Y ss (L(χω )) 6= ∅ if and only if ωi ≤ n1 i ki ωi for all
P
1 ≤ i ≤ m if and only if max{ωi }i ≤ n1 i ki ωi ;
P
(2) Y s (L(χω )) 6= ∅ if and only if ωi < n1 i ki ωi for all 1 ≤ i ≤ m if and
P
only if max{ωi }i < n1 i ki ωi .
The necessary parts of both (1) and (2) are true.
4.7.2. Diagonal hypersimplex and G-ample cone.
The previous conjectures
polyP
P lead to the discovery of the following
tope. Setting xi = nωi / i ki ωi , then xi satisfy 0 ≤ xi ≤ 1 and i ki xi = n.
Hence we introduce the polytope
X
∆m
ki xi = n.}.
n,{ki } = {(x1 , . . . , xm )|0 ≤ xi ≤ 1,
i
Recall the standard hypersimplex
∆m
n
is defined as
X
∆m
=
{(x
,
.
.
.
,
x
)|0
≤
x
≤
1,
xi = n.}.
1
m
i
n
i
k1 +···+km
. In fact, let Dk1 ,...,km be the diagThus ∆m
n,{ki } is a subpolytope of ∆n
k1 +···+km
onal subspace of R
such that the first k1 coordinates coincide, the
next k2 coordinate coincide, and so on, then
k1 +···+km
∩ Dk1 ,...,km .
∆m
n,{ki } = ∆n
m
Clearly ∆m
n,{ki } is the hypersimplex ∆n when all ki are equal to 1. Hence, it
seems reasonable to call ∆m
n,{ki } a diagonal hypersimplex or simply a generalized hypersimplex.
Let G = SL(V ) and G0 = S(GL(k1 , C) × · · · × GL(km , C)). Let also C G (X)
0
and C G (Y ) be the G-ample cone of X and G0 -ample cone of Y , respectively. (For the definition and properties of a general G-ample cone, see
Definition 3.2.1 and §3 of [17].)
74
Then, as a corollary of either of the above conjectures, we have
0
Conjecture 4.7.4. Both C G (X) and C G (Y ) can be naturally identified with the
positive cone over the generalized hypersimplex ∆m
n,{ki } .
4.7.3. Walls and Chambers.
In §3 of [17], a natural wall and chamber structure in C G (X) is introduced.
However it can happen that there are no (top) chambers at all in C G (X).
Not many examples of this type are previously known. Here we produce
an interesting one.
Consider the product of m-copies of Gr(2, C4 ),
4
X = Πm
i=1 Gr(2, C ).
4
Proposition 4.7.5. All the above conjectures are true for Πm
i=1 Gr(2, C ).
Proof. We only need to prove it for Conjecture 4.7.2, the rest follow from
4
this. Take any configuration {Vi } ∈ Πm
i=1 Gr(2, C ) such that Vi ∩ Vj =
{0} (iP6= j). We willPcheck that {Vi } is ℘ω -semistable. First note that
1
1
i ωi dim Vi = 2
i ωi . Let F be an arbitrary proper subspace of V .
dim V
We examine it case by case.
dim F = 1. F can intersect non-trivially (i.e., be contained in) only one
Vi . Hence we have
1 X
1X
ωi dim(F ∩ Vi ) ≤ ωi ≤
ωi .
dim F i
2 i
dim F = 2. If dim F ∩ Vi = 2, then F = Vi , hence
1 X
1X
ωi dim(F ∩ Vi ) = ωi ≤
ωi .
dim F i
2 i
Otherwise, dim F ∩ Vi ≤ 1 for all i, hence
1 X
1X
ωi dim(F ∩ Vi ) ≤
ωi .
dim F i
2 i
dim F = 3. If dim F ∩ Vi ≤ 1 for all i, then the stability condition is
trivially true. Otherwise, dim F ∩ Vi = 2 can only be true for only one i. In
this case,
X
1 X
1
ωi dim(F ∩ Vi ) ≤ (2ωi +
ωj )
dim F i
3
j6=i
X
1
1X
= (ωi +
ωj ) ≤
ωj .
3
2 j
j
¤
75
Proposition 4.7.6. For every weight set ω ∈ ∆m
4,{2,··· ,2} ,
X ss (Lω ) \ X s (Lω ) 6= ∅.
In particular, there is not any (top) chamber in the G-ample cone.
Proof. Let F be a 2-dimensional subspace. Take a configuration {Vi } ∈
4
Πm
i=1 Gr(2, C ) such that Vi ∩ Vj = {0} (i 6= j) and dim Vi ∩ F = 1 for all i.
Then {Vi } is semistable for all ω ∈ ∆m
4,{2,··· ,2} by the proof of the previous
proposition. Since
1 X
1X
1 X
ωi dim(F ∩ Vi ) =
ωi =
ωi dim Vi ,
dim F i
2 i
dim V i
{Vi } ∈ X ss (Lω ) \ X s (Lω ) for all
ω ∈ ∆m
4,{2,··· ,2} .
¤
Remark 4.7.7. Finally, note that
∆m
4,{2,··· ,2}
= {(x1 , . . . , xm )|0 ≤ xi ≤ 1,
m
X
2xi = 4}
i=1
= {(x1 , . . . , xm )|0 ≤ xi ≤ 1,
m
X
xi = 2}
i=1
which is just the standard hypersimplex ∆m
2 . Recall we just showed that
has
no
chambers
as
SL(4,
C)-ample
cone of Gr(2, C4 )m . However,
∆m
4,{2,··· ,2}
1 m
∆m
2 , as SL(2, C)-ample cone of (P ) has natural wall and chamber structure. It would be interesting to investigate in detail the implications of the
above on the problem of variation of GIT quotients by the two distinct,
yet related actions. Likewise, one should also study the implication of the
identity
m
∆m
kn,{n,...,n} = ∆k .
76
5. S TRATIFICATION OF THE S ET OF U NSTABLE P OINTS
Fix a linearization L and abbreviate X ss (L) by X ss . In this chapter, we
will study a canonical stratification of X with X ss as the unique open stratum. This is the same to give a stratification of X us .
This stratification can be used to calculate the cohomology of X ss (L)//G.
More importantly, we will use it to study variation of GIT quotients.
5.1. The function M L (x) and the numerical criterion revisited.
Let σ : G × X → X be an algebraic action of a connected reductive
linear algebraic group G on an irreducible algebraic variety X. Let L be a
G-linearized line bundle over X.
We will now re-examine the Hilbert-Mumford numerical criterion from
a different perspective.
Assume L is ample. Replacing L with some positive power L⊗n , we may
assume that L is very ample. Choose a G-equivariant embedding of X in
a projective space P(V ) such that G acts on X via its linear representation
in V .
5.1.1. The torus case.
To start with, let us assume that G is the n-dimensional torus (C∗ )n .
Then its group of characters X (G) is isomorphic to Zn . The isomorphism
is defined by assigning to any (m1 , . . . , mn ) ∈ Zn the homomorphism χ :
G → C∗ defined by the formula
mn
1
χ((t1 , . . . , tn )) = tm
1 · . . . · tn .
Any one-parameter subgroup λ : C∗ → G is given by the formula
λ(t) = (tr1 , . . . , trn )
for some (r1 , . . . , rn ) ∈ Zn . In this way we can identify the set X∗ (G) with
the group Zn . Let λ ∈ X∗ (G) and χ ∈ X (G); the composition χ ◦ λ is a
homomorphism C∗ → C∗ , hence is defined by an integer m. We denote
this integer by hλ, χi. It is clear that the pairing
X∗ (G) × X (G) → Z,
(λ, χ) 7→ hλ, χi
is isomorphic to the natural dot-product pairing
Zn × Zn → Z.
Now let
V =
M
χ∈st(V )
77
Vχ ,
where Vχ = {v ∈ V : g · v = χ(g) · v}. Here st(V ) is an index subset in X (G)
such that the eigenspace Vχ is nonzero.
P
For any v ∈ V we can write v = χ vχ , where vχ ∈ Vχ . The group G acts
on a vector v by the formula
X
g·v =
χ(g) · v
χ
Let x ∈ P(V ) be represented by a vector v in V . We set
st(x) = {χ : vχ 6= 0}
(the state set of x)
st(x) = convex hull of st(x) in X (G) ⊗ R ∼
= Rn .
Then
µL (x, λ) = min hλ, χi.
χ∈st(x)
In particular,
x ∈ X ss (L) ⇔ 0 ∈ st(x)
x ∈ X s (L) ⇔ 0 ∈ the interior of st(x).
(see Exercise §3.2 (2).)
In fact, the stability or instability of the point x can be measured by
signed distance from the origin to the boundary of the polytope st(x). To
do this, we use the Euclidean norm k • k on Rn . Then
minχ∈st(x) hλ, χi
µL (x, λ)
=
kλk
kλk
is equal to the signed distance from the origin to the boundary of the projection of st(x) to the positive ray spanned by the vector λ. And,
µL (x, λ)
M (x) = supλ∈X∗ (G)
kλk
L
is equal to the signed distance from the origin to the boundary of st(x).
One can restate the numerical criterion using the function M L (x): For
any ample linearized line bundle L
X ss (L) = {x ∈ X : M L (x) ≤ 0},
X s (L) = {x ∈ X : M L (x) < 0}.
78
5.1.2. The general reductive case.
Now if G is any reductive group we can fix a maximal torus T in G
and let W = NG (T )/T be its Weyl group. We denote by X∗ (G) the set of
one-parameter subgroups of G. We have
[
X∗ (G) =
X∗ (gT g −1 ).
g∈G
Let us identify X∗ (T ) ⊗ R with Rn and consider a W -invariant Euclidean
norm k k in Rn . Then we can define for any λ ∈ X∗ (G),
kλk := kInt(g) ◦ λk,
where Int(g) is the inner automorphism of G such that Int(g) ◦ λ ∈ X∗ (T ).
Definition 5.1.1. For any linearized line bundle L (not necessarily ample),
set
µL (x, λ)
L
µ̄ (x, λ) :=
kλk
L
M (x) = supλ∈X∗ (G) µ̄L (x, λ).
We shall show that M L (x) is always finite.
Lemma 5.1.2. Assume L is ample. Let T be a maximal torus of G and
P icG (X) → P icT (X)
L → LT
be the restriction map. Then for any x ∈ X, the set {M LT (g · x), g ∈ G} is finite
and M L (x) = maxg∈G M LT (g · x).
Proof. By §5.1.1, M LT (g · x) is the signed distance to the boundary of the
polytope st(g · x). The vertices of st(g · x) are among the characters of T in
st(V ) which is a finite set, hence there are only finitely many polytopes of
the form st(g · x). This shows that {M LT (g · x), g ∈ G} is finite.
That M L (x) = maxg∈G M LT (g · x) follows from the fact that
µL (g · x, λ) = µ(x, g −1 λg)
for any g ∈ G.
¤
Proposition 5.1.3. For any L ∈ P icG (X) (not necessarily ample), M L (x) is
finite.
Proof. It follows from the previous discussion that M L (x) is finite if L is
ample. It is known that for any L ∈ P ic(X) and an ample L1 ∈ P ic(X)
the bundle L ⊗ L⊗N
is ample for sufficiently large N . This shows that any
1
79
L ∈ P icG (X) can be written as a difference L1 ⊗ L−1
2 for ample L1 , L2 ∈
P icG (X). We have
−1
M
L−1
2
−µL2 (x, λ)
µL2 (x, λ)
µL2 (x, λ)
(x) = sup
= sup
= − inf
.
λ
kλk
kλk
kλk
λ
λ
If G is a torus, then it follows from the last paragraph of §5.1.1 that
L2 (x,λ)
λ
inf λ µ kλk
is finite. One can also argue as follows. The function kλk
→
µL2 (x,λ)
kλk
extends to a continuous function on the sphere of radius 1. Hence
it is bounded from below and from above.
If G is any reductive group, and T its maximal torus, let rT : P icG (X) →
L2 (x,λ)
=
P icT (X) be the restriction homomorphism. Then we have that µ kλk
µrT (L2 ) (g·x,λ1 )
kλ1 k
for some g ∈ G and some λ1 ∈ X∗ (T ). Since the set of possible
state sets stT (g · x) is finite, we obtain that
inf
λ
µrT (L2 ) (g · x, λ1 )
µL2 (x, λ)
= inf inf
g λ1
kλk
kλ1 k
is finite. Now
−1
−1
µL1 (x, λ)
µL2 (x, λ)
µL1 (x, λ) µL2 (x, λ)
+
) ≤ sup
+ sup
M (x) = sup(
kλk
kλk
kλk
kλk
λ
λ
λ
L
which implies that M L (x) is finite because M L (x) is obviously bounded
from below.
¤
One can restate the numerical criterion using the function M L (x):
Theorem 5.1.4. For any ample L ∈ P icG (X), we have
X ss (L) = {x ∈ X : M L (x) ≤ 0},
X s (L) = {x ∈ X : M L (x) < 0}.
80
5.2. Adapted one-parameter subgroups.
For every one-parameter subgroup λ one defines a subgroup P (λ) ⊆ G
by
P (λ) := {g ∈ G : lim λ(t) · g · λ(t)−1 exists in G}.
t→0
This is a parabolic subgroup of G; λ is contained in its radical. The set
L(λ) := {ḡ = lim λ(t) · g · λ(t)−1 : g ∈ P (λ)}
t→0
is a subgroup of P (λ) which centralizes λ, the set of g’s such that the limit
equals 1 form the unipotent radical U (λ) of P (λ) (see [MFK], Prop. 2.6).
We have
P (λ) = U (λ) × L(λ).
Let
M
Lie(G) = t ⊕
Lie(G)α
α∈Φ
be the root decomposition for the Lie algebra of G, where t = Lie(T ). Then
M
M
Lie(L(λ)) = t ⊕
Lie(G)α , Lie(U (λ)) =
Lie(G)α .
hλ,αi=0
hλ,αi>0
Definition 5.2.1. 1.2.2 Definition. Let L ∈ P icG (X), x ∈ X. A one-parameter
subgroup λ is called adapted to x with respect to L if
M L (x) =
µL (x, λ)
.
kλk
The set of primitive (i.e. not divisible by any positive integer) adapted
one-parameter subgroups will be denoted by ΛL (x).
Corollary 5.2.2. 1.2.3 Corollary. Assume L is ample. The set ΛL (x) ∩ X∗ (T ) is
non-empty and finite.
Proof. By Lemma 5.1.2, ΛL (x) is maxg∈G M LT (g · x) and {M LT (g · x)|g ∈ G}
is a finite set. Hence there is a particular g such that ΛL (x) = M LT (g · x).
This follows from observing that the set of all possible states st(x) with
respect to rT (L) is finite. .
¤
Theorem 5.2.3. 1.2.4 Theorem. Assume L is ample and x ∈ X us (L). Then
(i) There exists a parabolic subgroup P (L)x ⊆ G such that P (L)x = P (λ) for all
λ ∈ Λ(L)x .
(ii) All elements of ΛL (x) are conjugate to each other by elements from P (λ).
(iii) If T is a maximal torus contained in P (L)x , then ΛL (x) ∩ X∗ (T ) forms one
orbit with respect to the action of the Weyl group.
Proof. This is a theorem of G. Kempf [Ke] .
81
¤
Corollary 5.2.4. 1.2.5 Corollary. Assume x ∈ X us (L). Then for all g ∈ G
(i) Λ(gx) = Int(g)Λ(x);
(ii) P (L)g·x = Int(g)P (L)x ;
(iii) Gx ⊂ P (L)x .
Theorem 5.2.5. 1.2.6 Theorem. Assume L ∈ P icG (X) is ample and x ∈
X us (L). Let λ ∈ Λ(L)x and y = limt→0 λ(t) · x. Then
(i) λ ∈ Λ(L)y ;
(ii) M L (x) = M L (y);
Proof. This is Theorem 9.3 from [Ne2].
¤
Theorem 5.2.6. 2.1.9 Theorem. For any x ∈ X, M L (x) is equal to the signed
distance from the origin to the boundary of Φ(G · x). If x ∈ X us (L), then the
following three properties are equivalent:
(i) M L (x) = kΦ(x)k;
(ii) x is a critical point for ||Φ||2 with nonzero critical value;
(iii) for all t ∈ R, exp(tΦ∗ (x)) ∈ Gx , and the complexification of exp(RΦ∗ (x)) is
adapted for M L (x).
82
5.3. Stratification of the set of unstable points. . Following Hesselink
[He], we introduce the following stratification of the set X us (L).
1.3.1 For each d > 0 and the conjugacy class < τ > of a one-parameter
subgroup τ of G, we set
L
L
Sd,<τ
> = {x ∈ X : M (x) = d, ∃g ∈ G
such that Int(g) ◦ τ ∈ Λ(L)x }.
Let T be the set of conjugacy classes of one-parameter subgroups. Hesselink shows that for any ample L
[
L
X = X ss (L) ∪
Sd,<τ
>
d>0,<τ >∈T
is a finite stratification of X into Zariski locally closed G-invariant subvarieties of X. Observe that property (ii) of Theorem 1.2.4 ensures that the
subsets Sd,<τ > , d > 0, are disjoint.
L
1.3.2 Let x ∈ Sd,<τ
> . For each λ ∈ Λ(x) we can define the point y =
L
limt→0 λ(t) · x. By Theorem 1.2.6, y ∈ Sd,<τ
> and λ fixes y. Set
L
L
∗
Zd,<τ
> := {x ∈ Sd,<τ > : λ(C ) ⊂ Gx
for some λ ∈< τ >}.
L
Let us further subdivide each stratum Sd,<τ
> by putting for each λ ∈< τ >
L
L
L
Sd,λ
:= {x ∈ Sd,<τ
> : λ ∈ Λ (x)}.
Hesselink calls these subsets blades. If
L
L
Zd,λ
:= {x ∈ Sd,λ
: λ(C∗ ) ⊂ Gx },
then we have the map
L
L
pd,λ : Sd,λ
→ Zd,λ
, x 7→ y = lim λ(t) · x.
t→0
Proposition 5.3.1. 1.3.3 Proposition.
L
L
L
(i)Sd,λ
= {x ∈ X : limt→0 λ(t) · x ∈ Zd,λ
} = p−1
d,λ (Zd,λ );
L
L
(ii) for each connected component Zd,λ,i
of Zd,λ
the restriction of the map pd,λ :
L
L
L
L
Sd,λ
→ Zd,λ
over Zd,λ,i
is a vector bundle with the zero section equal to Zd,λ,i
,
assuming in addition that X is smooth;
L
L
is L(λ)-invariant; if ḡ denotes the projection of
is P (λ) -invariant, Zd,λ,i
(iii)Sd,λ
L
g ∈ P (λ) to L(λ), then for each x ∈ Sd,λ
, pd,λ (g · x) = ḡ · pd,λ (x);
L
L
(iv) there is a surjective finite morphism G ×P (λ) Sd,λ
→ Sd,<τ
> . It is bijective if
L
d > 0 and is an isomorphism if Sd,<τ > is normal.
Proof. See [Ki1], §13.2, 13.5.
¤
L
⊂ X λ . As usual
1.3.4 Let X λ = {x ∈ X : λ(C∗ ) ⊂ Gx }. By definition, Zd,λ
we may assume that G acts on X via its linear representation in a space
V = Γ(X, L)∗ . Let V = ⊕i Vi , where Vi = {v ∈ V : λ(t)·v = ti v}. Then X λ =
83
L
∪i Xiλ , where Xiλ = P(Vi ) ∩ X. For any x ∈ Zd,λ
, d = M L (x) = µL (x, λ).
If v represents x in V , then, by definition of µL (x, λ), we have v ∈ Vd .
L
Therefore, Zd,λ
⊂ Xdλ . It easily follows from the definition of P (λ) that L(λ)
leaves each subspace Vi invariant. By Proposition 1.3.3 (iii), the group L(λ)
L
leaves Zd,λ
invariant. The central subgroup λ(C∗ ) of L(λ) acts identically
on P(Vd ) , hence one defines the action of L(λ)0 = L(λ)/λ(C∗ ) on P(Vd )
L
leaving Zd,λ
invariant. Let OXdλ (1) be the very ample L(λ)0 -linearized line
bundle obtained from the embedding of Xdλ into P(Vd ).
L
= (Xdλ )ss (OXdλ (1)).
Proposition 5.3.2. 1.3.5 Proposition. Zd,λ
Proof. See [Ne2],Theorem 9.4.
¤
84
5.4. Finiteness theorems.
Proposition 5.4.1. 1.3.6 Lemma. Let Π(X λ ) be the set of connected components
of X λ , λ ∈ X∗ (G). Then the group G acts on the union of the sets Π(X λ ) (λ ∈
X∗ (G)) via its action on the set X∗ (G) with finitely many orbits.
Proof. Choose some G-equivariant projective embedding of X ,→ P(V ).
Let T be a maximal torus of G and V = ⊕χ Vχ be the decomposition into
the direct sum of eigenspaces with respect to the action of T . Since any λ
is conjugate to some one-parameter subgroup of T , it is enough to show
that the set (∪λ∈X∗ (T ) Π(X λ )) is a finite union. For each integer i and λ, let
Vi,λ = ⊕hχ,λi=i Vχ . Then X λ = ∪i P(Vi,λ ) ∩ X. Since the number of nontrivial direct summands Vχ of V is finite, there are only finitely many different subvarieties of the form X λ . Each of them consists of finitely many
connected components.
¤
1.3.7 For each connected component Xiλ of X λ , one can define its contracting set Xi+ = {x ∈ X : limt→0 λ(t) · x ∈ Xiλ }. By a theorem of BialynickiBirula [B-B2], this set has the structure of a vector bundle with respect
to the natural map x 7→ limt→0 λ(t) · x when X is smooth. The decomposition X = ∪i Xi+ is the so-called Bialynicki-Birula decomposition induced by λ. Although there are infinitely many one-parameter subgroups
T ⊂ G, the number of their corresponding Bialynicki-Birula decompositions is finite. In fact, there is a decomposition of X∗ (T ) into a finite union
of rational cones such that two one-parameter subgroups give rise to the
same Bialynicki-Birula decomposition if and only if they lie in the same
cone. This is Theorem 3.5 of [Hu2]. A simple example of this theorem is
the case of the flag variety G/B acted on by a maximal torus T . In this
case, the cone decompostion of X∗ (T ) is just the fan formed by the Weyl
chambers. Each Weyl chamber gives a Bruhat decomposition (there are
|W | of them, where W is the Weyl group). Other Bialynicki-Birula decompositions correspond to the faces of the Weyl chambers. The following
proposition follows from Theorem 3.5 of [Hu2].
Proposition 5.4.2. 1.3.8 Proposition. Let T be a fixed maximal torus of G. Then
there are only finitely many Bialynicki-Birula decompositions induced by λ ∈
X∗ (T ).
In fact this proposition can also be proved as follows without appealing to
the cone decomposition of X∗ (T ). Let W = ∪i Wi be the union of the connected components of the fixed point set of T . Two points x and y of X are
called equivalent if whenever T · x ∩ Wi 6= ∅, then T · y ∩ Wi 6= ∅, and vice
versa. This gives the decomposition of X into a finite union of equivalence
classes X = ∪E X E , where E ranges over all equivalence classes (X E are
called torus strata in [Hu2]). It is proved in Lemma 6.1 of [Hu2] that every
85
Bialynicki-Birula stratum Xi+ is a (finite) union of X E . This implies that
there are only finitely many Bialynicki-Birula decompositions.
Theorem 5.4.3. 1.3.9 Theorem.
(i) The set of locally closed subvarieties S of X which can be realized as the stratum
L
G
Sd,<τ
> for some ample L ∈ P ic (X), d > 0 and τ ∈ X∗ (G) is finite.
(ii) The set of possible open subsets of X which can be realized as a subset of
semi-stable points with respect to some ample G-linearized line bundle is finite.
Proof. The second assertion obviously follows from the first one. We prove
the first assertion by using induction on the rank of G. The assertion is
obvious if G = {1}. Let us apply the induction to the case when X is
equal to a connected component Xiλ of X λ (λ ∈ X∗ (T )) and G = L(λ)/Ti
where Ti is the isotropy of Xiλ in T . It is well known that only finitely
many subgroups of T can realized as isotropy subgroups. Thus we obtain
that the set of open subsets of Xiλ which can be realized as the sets of
semi-stable points of some ample L(λ)/Ti -linearized line bundle is finite.
By Proposition 1.3.5 and Lemma 1.3.6, this implies that the set of locally
L
closed subsets of X which can be realized as the subsets Zd,<τ
> is finite.
Now, by Proposition 1.3.3 (i) and Proposition 1.3.8, out of the finitely many
L
L
Zd,<τ
¤
> , one can only construct finitely many Sd,<τ > , and we are done.
Exercises
86
6. M OMENT M APS
AND
GIT
6.1. Quantization commutes with reduction.
Consider a very ample linearization L and an induced equivariant embedding X ⊂ P(V ) where V = H 0 (X, L)∗ . Let
M
V = V0 ⊕
Vχ
χ
be a decomposition of V into direct sum of trivial G-module V0 and irreducible ones Vχ with χ 6= 0 the highest weight (so, Vχ may repeat).
Proposition 6.1.1. The L very ample line bundle L descends to a very ample line
bundle M over the quotient X ss (L)//G such that
H 0 (X ss (L)//G, M ) = H 0 (X, L)G ∼
= V ∗.
0
Proof. Choose a basis {s0 , · · · , sr } of V0∗ which identified with H 0 (X, L)G .
By the proof of Theorem 2.7.1, [s0 : · · · : sr ] defines an embedding of
X ss (L)//G into P(V0 ). Clearly L descends to the ample line bundle M =
O(X ss (L)). It follows immediately that
H 0 (X ss (L)//G, M ) = V0∗ = H 0 (X, L)G .
¤
Remark 6.1.2. ????? Do this for CHOW QUOTIENT
87
6.2. GIT quotients of X × G/B and general symplectic reductions.
Let G be a complex reductive algebraic group, K a maximal compact
subgroup. Let T be a maximal compact torus in K so that its complexification is a maximal complex torus in G. The Lie algebras of G, K, H and
T are denoted g, k, h and t, respectively.
Via derivations any character χ : T → U (1) of T is identified with a
linear function on t. The set of such functions is a lattice t∗Z in t∗ . Each
coadjoint orbit of K in k∗ intersects t∗ at a unique orbit of the Weyl group
acting on t∗ . If we fix a positive Weyl chamber t∗+ , then we obtain a bijective
correspondence between coadjoint orbits and points of t∗+ .
For any integral point α ∈ t∗Z , it defines a character of T and hence also
a character of H. The character as an integral point in the weight lattice,
defines an irreducible G-module Vα . By the Borel-Weil theorem, it defines
a very ample line bundle Lα over G/B such that
Γ(G/B, Lα ) = Vα∗ = V−w0 α
where w0 is the longest element in the Weyl group. A choice of basis of
Vα∗ embeds G/B into P(Vα ). Choose a principal weight vector vα ∈ Vα , the
image of G/B in P(Vα ) is the K-orbit through [vα ], and hence it can also be
identified with the coadjoint orbit Oα through α. One can check from the
definition that
Lemma 6.2.1. The moment map Oα ∼
= G/B → k∗ is simply the inclusion map.
Let Ōp denote the symplectic manifold obtained from the symplectic
manifold Op by replacing its symplectic form ω by −ω. Then the product symplectic manifold X × Ōp admits a moment map Φp defined by the
formula
Φp (x, q) = Φ(x) − q.
−1
p
Now the set Φ−1
p (0) becomes identified with the set Φ (O ) and
−1
−1
p
Φ−1
p (0)/K = Φ (p)/Kp = Φ (O )/K
where Kp is the isotropy subgroup of K in K at p. This quotient space
is called the Marsden-Weinstein reduction or symplectic reduction of X with
respect to p. Evidently it depends only on the orbit of p.
The following result (see [Ne2], Appendix by D. Mumford) is another
interpretation of Theorem 2.1.8.
Theorem 6.2.2. 2.2.4 Theorem. Let L be a very ample G-linearized line bundle
on X; L( χn ) denotes the line bundle on X × G/B equal to the tensor product of
the preimages of the bundles L⊗n and Lχ under the projection maps. Then L( χn )
defines a G-equivariant embedding of X × G/B into a projective space and hence
88
a moment map Φ nχ : X × G/B → Lie(K)∗ . We have
∼ −1 (O nχ ).
Φ−1
χ (0) = Φ
n
In particular,
χ
χ
−1
(X × G/B)ss (L( ))//G = Φ−1
(O n )/K.
χ (0)/K = Φ
n
n
89
6.3. Rational points in the moment map image.
We now describe the image Φ(X) ⊂ k∗ . We can use some non-degenerate
K-invariant quadratic form on k to identify k∗ with k (in the case that K is
semisimple, one simply uses the Killing form). This changes the moment
map to a map Φ∗ : X → k. Let h be a Cartan algebra of k and let H be the
maximal torus in K whose Lie algebra is Lie(H) = h. Via derivations any
character χ : H → U (1) of H is identified with a linear function on h. The
set of such functions is a lattice h∗Z in h∗ . Under the isomorphism h → h∗
defined by the non-degenerate quadratic form, we obtain a Q-vector subspace hQ of h which defines a rational structure on h. It is known that each
adjoint orbit intersects h at a unique orbit of the Weyl group acting on h. If
we fix a positive Weyl chamber h+ , then we obtain a bijective correspondence between adjoint orbits and points of h+ . This defines the reduced
moment map:
\
Φred : X → h+ , x 7→ Φ∗ (Kx) h+ .
A co-adjoint orbit O is called rational if under the correspondence
co-adjoint orbits ↔ h+
it is defined by an element α in hQ . We can write α in the form χn for some
integer n and χ ∈ X (T ), where T is a fixed maximal torus of G. Elements
of h+ of the form χn will be called rational. We denote the set of such elements by (h+ )Q . By the Borel-Weil theorem, χ determines an irreducible
representation V (χ) which can be realized in the space of sections of an
ample line bundle Lχ on the homogeneous space G/B, where B is a Borel
subgroup containing T . Its highest weight is equal to χ.
Theorem 6.3.1. Let Φred : P(V ) → h+ be the reduced moment map for the action
of K on P(V ). Then Φred (X) is a compact convex subpolytope of Φred (P(V )) with
vertices at rational points. Moreover
\
χ
Φred (X) (h+ )Q = { : V (χ)∗ is a direct
n
summand of Γ(X, L⊗n ) as a G-module }.
This result is due to D. Mumford (see the proof in [Ne2], Appendix, or
in [Br]). Note that the assertion is clear in the situation of 1.1.5. In that case
the image of the moment map is equal to the convex hull Conv(St(V )) of
the set St(V ) = {χ : Vχ 6= {0}}. This convexity result was first observed by
M. Atiyah [At] in the setting of symplectic and Kähler geometry.
6.4. The function M L (x) and moment map.
Theorem 6.4.1. For any x ∈ X, M L (x) is equal to the signed distance from the
origin to the boundary of Φ(G · x).
90
6.5. Homological equivalence for G-linearized line bundles.
We as-
sume X to be a projective variety.
2.3.1. Recall the definition of the Picard variety P ic(X)0 and the NéronSeveri group N S(X). We consider the Chern class map c1 : P ic(X) →
H 2 (X, Z) and put
P ic(X)0 = Ker(c1 ), N S(X) = Im(c1 ).
One way to define c1 is to choose a Hermitian metric on L ∈ P ic(X), and
set c1 (L) to be equal to the cohomology class of the curvature form of this
metric. Thus elements of P ic(X)0 are isomorphism classes of line bundles which admit a Hermitian form with exact curvature form Θ. If Θ is
given locally as 2πi d0 d00 log(ρU ), then Θ is exact if and only if there exists a
global function ρ(x) such that d0 d00 log(ρU /ρ) = 0 for each U . This implies
that ρU /ρ = |φU | for some holomorphic function φU on U . Replacing the
0
transition functions σU V of L by φU σU φ−1
V we find an isomorphic bundle L
with transition functions σU0 V with |σU0 V | = 1. Since σU0 V are holomorphic,
we obtain that the transition functions of L0 are constants of absolute value
1. The Hermitian structure on L0 is given by a global function ρ.
2.3.2. Now assume L is a G-linearized line bundle, where G is a complex Lie group acting holomorphically on X. Let K be its compact real
form. By averaging over X (here we use that K and X are compact)
we can find a Hermitian structure on L such that K acts on L preserving this structure (i.e. the maps Lx → Lgx are unitary maps). The curvature form of L with respect to the K-invariant Hermitian metric is a
K-invariant 2-form Θ of type (1,1). If L is ample, Θ is a Kähler form
and its imaginary part is a K-invariant symplectic form on X. A different choice of K-invariant Hermitian structure on L replaces Θ with
Θ0 = Θ + 2πi d0 d00 log(ρ) where ρ is a K-invariant positive-valued smooth
real function on X. This can be seen as follows. Obviously Θ − Θ0 is
a K-invariant 2-form d0 d00 logρ for some global function ρ. By the invariance, we get for any g ∈ K, d0 d00 (log(ρ(gx)/ρ(x)) = 0. This implies that
ρ(gx)/ρ(x) = |cg (x)| for some holomorphic function cg (x) on X. Since X is
compact, this function must be constant, and the map g 7→ |cg | is a continuous homomorphism from K to R∗ . Since K is compact and connected, it
must be trivial. This gives |cg | = ρ(gx)/ρ(x) = 1, hence ρ(x) is K-invariant.
Taking the cohomology class of Θ we get a homomorphism:
c : P icG (X) → H 2 (X, Z).
Proposition 6.5.1. 2.3.3 Proposition. There is a natural isomorphism:
Ker(c) ∼
= X (G) × P ic(X)0 .
Proof. It is known (see [KKV]) that the cokernel of the canonical forgetful
homomorphism P icG (X) → P ic(X) is isomorphic to P ic(G). Since the
91
latter group is finite, it is easy to see that that Ker(c) is mapped surjectively
to P ic(X)0 with the kernel isomorphic to the group X (G) of characters of
G. To prove the assertion it suffices to construct a section s : P ic(X)0 →
P icG (X). Now let L ∈ P ic(X)0 . We can find an open trivializing cover
{Ui }i∈I of L such that L is defined by constant transition functions σU V .
For each g ∈ G the cover {g(Ui )}i∈I has the same property. Moreover,
σg(U )g(V ) = (g −1 )∗ (σU V ) = σU V . This shows that we can define the (trivial)
action of G on L by the formula g · (x, t) → (g · x, t) which is well-defined
and is a G-linearization on L. This allows one to define the section s and
check that s(P ic(X)0 ) ∩ X (G) = {1}.
¤
Definition 6.5.2. 2.3.4 Definition. A G-linearized line bundle L is called
homologically trivial if L ∈ Ker(c) and the image of L in X (G) is the identity. In other words, L is homologically trivial if it belongs to P ic(X)0
as a line bundle, and the G-linearization on L is trivial (in the sense of
the previous proof). The subgroup of P icG (X) formed by homologically
trivial G-linearized line bundles is denoted by P icG (X)0 . Two elements of
P icG (X) defining the same element of the factor group P icG (X)/P icG (X)0
are called homologically equivalent.
Lemma 6.5.3. 2.3.5 Lemma. Let L and L0 be two homologically equivalent Glinearized line bundles. Then for any x ∈ X and any one-parameter subgroup
λ
0
µL (x, λ) = µL (x, λ).
Proof. It is enough to verify that for any homologically trivial G-linearized
line bundle L and any one-parameter subgroup λ we have µL (x, λ) = 1.
But this follows immediately from the definition of the trivial G-linearization
on L.
¤
Theorem 6.5.4. 2.3.6 Theorem. Let L and L0 be two ample G-linearized bundles.
Suppose that L is homologically equivalent to L0 . Then the moment maps defined
by the corresponding Fubini-Study symplectic forms differ by a K-invariant constant.
Proof. Choose some symplectic curvature forms ω and ω 0 of L and L0 , respectively. By assumption, ω 0 = ω + 2πi d0 d00 log(ρ) for some positive-valued
K-invariant function ρ (see 2.3.2). Thus we can write ω = ω 0 + dθ, where
θ is a K-invariant 1-form of type (0, 1). By definition of the moment map,
we have for each ξ ∈ Lie(K)
d(ξ ◦ Φ0 ) = ıω0 (ξ ] ) = ıω+dθ (ξ ] ) = d(ξ ◦ Φ) + L(ξ ] )θ − dhξ ] , θi,
where L(ξ ] ) denotes the Lie derivative along the vector field ξ ] . Since θ is
K-invariant, we get L(ξ ] )θ = 0. Because G (hence also K) acts on X holomorphically, each element of Lie(G) defines a holomorphic vector field. In
particular, ξ ] is holomorphic. But θ is of type (0, 1), so hξ ] , θi = 0. This
92
shows that d(ξ ◦ Φ0 ) = d(ξ ◦ Φ) and hence Φ0 − Φ is an ad(K)-invariant
constant in Lie(K)∗ .
¤
2.3.7 Let N S G (X) = P icG (X)/P icG (X)0 . By Proposition 2.3.3, we have an
exact sequence
0 → X (G) → N S G (X) → N S(X) → A,
where A is a finite group. It is known that the Néron-Severi group N S(X)
is a finitely generated abelian group. Its rank is called the Picard number
of X and is denoted by ρ(X). From this we infer that N S G (X) is a finitely
generated abelian group of rank ρG (X) equal to ρ(X) + t(G), where t(G)
is the dimension of the radical R(G) of G. An element of N S G (X) will be
called G-ample if it can be represented by an G-effective ample line bundle.
By Proposition 2.3.5, all ample G-linearized line bundles in the same Gample homological equivalence class are G-effective.
Let N S G (X)+ denote the subset of G-ample homological equivalence
classes. One checks that it is a semigroup in N S G (X). Let
N S G (X)R = N S G (X) ⊗ R
be the finite-dimensional real vector space generated by N S G (X).
2.3.8 One can give the following interpretation of elements of N S G (X)R .
First of all the space N S G (X)R contains the lattice N umG (X) isomorphic
to N S G (X)/Torsion. Secondly, it contains the Q-vector space N S G (X)Q .
For each l ∈ N S G (X)Q there is a smallest integer n such that nl is equal to
the class [L] of a G-linearized line bundle L in N umG (X). By using 2.3.2
and Theorem 2.3.6 we can interpret elements of N S G (X)R as the moment
maps given by all symplectic forms which arise as the imaginary parts of
a closed real K-invariant 2-forms of type (1,1).
2.4 Stratification of the set of unstable points via moment map. Here we
shall compare the Hesselink stratification of the set X \ X ss (L) with the
Ness-Kirwan stratification of the same set using the Morse theory for the
moment map. The main references here are [Ki1], [Ne2]
2.4.1. Let X be a compact symplectic manifold and K be a compact Lie
group acting symplectically on it with moment map Φ : X → Lie(K)∗ . We
choose a K-invariant inner product on Lie(K) and identify Lie(K) with
Lie(K)∗ . Let f (x) = kΦ(x)k2 . It is obvious that the critical points of the
map Φ are also the critical points of the function f : X → R. But some
minimal critical points of f need not be critical for Φ (consider the case
when 0 is a regular point of Φ). For any β ∈ Lie(K) we denote by Φβ the
composition of Φ and β : Lie(K)∗ → R. Let Zβ be the set of critical points
of Φβ with critical value equal to kβk. This is a symplectic submanifold of
X (possibly disconnected) fixed by the subtorus Tβ equal to the closure in
93
H of the real one-parameter subgroup exp Rβ, where H is the maximal
subtorus of K.
The composition of Φ and the natural map Lie(K)∗ → Lie(H)∗ is the
moment map ΦH for the action of H on X. If we use the inner product on
Lie(K) to identify Lie(H) with its dual, then ΦH (x) equals the orthogonal
projection of Φ(x) to Lie(H). Thus, if Φ(x) ∈ Lie(H) then ΦH (x) = Φ(x).
Hence such a point x is a critical point for f = kΦk2 if and only if it is
a critical point for the function fH = kΦH k2 . By a theorem of Atiyah [A],
ΦH (X) is equal to the convex hull of a finite set of points A in Lie(H). Each
point of A is the image of a fixed point of H. In general, the points of A are
not rational points in the vector space Lie(H).
Proposition 6.5.5. 2.4.2 Proposition.
(i) A point x ∈ X is critical for f if and only if the induced tangent vector Φ∗ (x)]x
at x equals 0.
(ii) Let x ∈ X be a point such that Φ(x) ∈ ~. Then x is a critical point for f if
and only if it is a critical point for fH .
(iii) x is a critical point for fH with critical value β 6= 0 if and only if x ∈ Zβ . In
this case β is the closest point to 0 of the convex hull of some subset of the set A.
Proof. This proposition follows from Lemmas 3.1, 3.3, and 3.12 from [Ki1].
¤
2.4.3. Fix a positive Weyl chamber ~+ and let B be the set of points β in ~+
which can be realized as the closest point to the origin of the convex hull
of some subset of the set A. For each β ∈ B we set
Cβ = K · (Zβ ∩ Φ−1 (β)).
Let Sβ (resp. Yβ ) denote the subset of points x ∈ X such that the limit set
of the path of steepest descent for f from x (with respect to some suitable
K-invariant metric on X) is contained in Cβ (resp. Zβ ). Then Yβ is a locally
closed submanifold of Sβ .
Theorem 6.5.6. 2.4.4 Theorem. The critical set of f is the disjoint union of the
closed subsets Cβ , β ∈ B. The sets {Sβ } form a smooth stratification of X.
Proof. This is Theorem 4.16 from [Ki1].
¤
2.4.5 Now assume that the symplectic structure on X is defined by the
imaginary part of a Kähler form on X. Let G be the complexification of K.
This is a reductive complex algebraic group. We assume that the action of
K on X is the restriction of an action of G on X which preserves the Kähler
structure of X. Then we have a stratification {Sβ } chosen with respect to
94
the Kähler metric on X. In this case we can describe the stratum Sβ as
follows:
Sβ = {x ∈ X : β is the unique closest point to 0 of Φred (G · x)}.
Let Zβmin denote the minimal Morse stratum of Zβ associated to the function kΦ − βk2 restricted to Zβ . Note that Φβ is the moment map for the
symplectic manifold Zβ with respect to the stabilizer group Stabβ of β under the adjoint action of K. Let Yβmin be the pre-image of Zβmin under the
natural map Yβ → Zβ .
For any β ∈ B let
P (β) = {g ∈ G : exp(itβ) · g · exp(−itβ) has a limit in G}.
It is a parabolic subgroup of G, and is the product of the Borel subgroup
of G and Stabβ which leaves Yβmin invariant.
Theorem 6.5.7. 2.4.6 Theorem.
(i) There is an isomorphism Sβ ∼
= G ×P (β) Y min ;
β
(ii) If x ∈
Yβmin
then {g ∈ G|gx ∈
Yβmin }
= P (β).
Proof. This is Theorem 6.18 and Lemma 6.15 from [Ki1].
¤
The relationship between the stratification {Sβ } and the stratification described in section 1.3 is as follows. First of all we need to assume that X is
a projective algebraic variety and the Kähler structure on X is given by the
curvature form of an ample L on X. In this case each β is a rational vector
in ~+ = Lie(H)+ . For any such β we can find a positive integer n such
that nβ is a primitive integral vector of ~+ and hence defines a primitive
one-parameter subgroup λβ of G.
Theorem 6.5.8. 2.4.7 Theorem. There is a bijective correspondence between the
L
moment map strata {Sβ }β6=0 and the Hesselink strata {Sd,<τ
> } given by Sβ →
L
Skβk,<λβ > . Under this correspondence
(i) P (β) = P (λβ ), Stabβ = L(λβ )(R);
λ
β
L
; Zβmin = Zkβk,λ
;
(ii)Zβ = Xkβk
β
L
(iii) Yβmin = Skβk,λ
.
β
Proof. See [Ki1], §12, or [Ne2].
¤
There is the following analog of Theorem 1.3.9:
Theorem 6.5.9. Theorem 2.4.8.
(i) The set of locally closed subvarieties S of X which can be realized as the stratum
Sβ for some K-equivariant Kähler symplectic structure and β ∈ Lie(H)∗ is finite.
95
(ii) The set of possible open subsets of X which can be realized as a stratum S0 for
some Kähler symplectic structure on X is finite
Proof. The second assertion obviously follows from the first one. We prove
the first assertion by using induction on the rank of K. The assertion is
obvious if K = {1}. Applying induction to the case when X is equal to a
connected component of Zβ and K = Stabβ /Tβ , we obtain that the set of
open subsets of some connected component of Zβ which can be realized
as the connected component of the set Zβmin is finite. The finiteness of the
set of subsets that can be realized as Zβ follows from Lemma 1.3.6. This
implies that the set of closed subsets of X which can be realized as the
subsets Zβmin is finite. Now each Zβmin determines Yβmin , and by Theorem
2.4.6, the stratum Sβ .
¤
2.5 Kähler quotients. We can extend many notions of GIT-quotients to the
case of symplectic reductions with respect to a moment map Φω : X →
Lie(K)∗ defined by a Kähler structure ω on X (see [Ki1]).
2.5.1 First we define
X ss (ω)c = G(Φω )−1 (0).
Assuming that K acts freely on (Φω )−1 (0), the symplectic reduction (Φω )−1 (0)/K
has a natural Kähler structure and is diffeomeorphic to the orbit space
X ss (ω)c /G.
2.5.2 For any x ∈ X and λ ∈ X∗ (G) we define
µω (x, λ) = kλkdλ (0, Φωred (G · x)),
where dλ (0, A) denotes the signed distance from the origin to the boundary
of the projection of the set A to the positive ray spanned by λ. Thus we can
define
M ω (x) = sup dλ (0, Φωred (G · x))
λ
ω
so that M (x) is equal to the distance from the origin to Φωred (G · x), and
set
X ss (ω) = {x ∈ X : M ω (x) ≤ 0},
X s (ω) = {x ∈ X : M ω (x) < 0},
and
X sss (ω) = X ss (ω) \ X s (ω).
As in the case of Hodge Kähler structures we can define the set of primitive
adapted one-parameter subgroups Λω (x). There are analogs of Theorems
1.2.4 and 1.2.6 in our situation.
2.5.3 The analog of the categorical quotient X ss //G is the orbit space (Φω )−1 (0)/K.
The analog of the geometric quotient is the orbit space X s (ω)∩(Φω )−1 (0)/K.
96
7. G- AMPLE CONE : WALLS
AND CHAMBERS
It can happens that there is not any chamber in the G-effective cone. A
trivial example is any case when Picard group is generated by one line
bundle and the group G is semisimple. For more interesting example,
consider the diagonal action of SL(4, C) on the product G(<)(2, C4 )n (n ≥
4), it can be shown that for this action chambers do not exist. The details
will presented in §??. The G-ample cone.
7.1. G-effective line bundles.
We will assume that X is projective and normal.
Definition 7.1.1. A G-linearized line bundle L on X is called G-effective
if X ss (L) 6= ∅. A G-effective ample G-linearized line bundle is called Gample.
Proposition 7.1.2. Let L be an ample G-linearized line bundle. The following
assertions are equivalent:
(i) L is G-effective;
(ii)Γ(X, L⊗n )G 6= {0} for some n > 0;
(iii) If Φ : X → Lie(K)∗ is the moment map associated to a G-equivariant
embedding of X into a projective space given by L, then Φ−1 (0) 6= ∅.
Proof. (i)⇔(ii). Follows from the definition of semi-stable points.
(ii) ⇔ (iii). Follows from Theorem 2.2.1.
¤
Proposition 7.1.3. 3.1.3 Proposition.
(i) If L and M are two G-effective G-linearized bundles then L⊗M is G-effective.
(ii) For any affine open G-invariant subset U of X, the restriction of a G-effective
line bundle L to U is a G-effective line bundle on U .
T
Proof. (i) Let x ∈ X ss (L) X ss (M ); then there exists s ∈ Γ(X, L⊗n )G and
s0 ∈ Γ(X, M ⊗m )G , for some n, m > 0 such that s(x) 6= 0, s0 (x) 6= 0 and Xs
and Xs0 are both affine. Without loss of generality we may assume that
n = m. Then the G-invariant section s ⊗ s0 of L⊗n ⊗ M ⊗n does not vanish at
x. Using the Segre embedding we get that Xs⊗s0 is affine. This shows that
x ∈ X ss (L ⊗ M ), hence L ⊗ M is G-effective.
(ii) Follows from the definition and the fact that intersection of affine open
sets is affine.
¤
Proposition 7.1.4. 3.1.4 Proposition. Let L, L0 be two ample G-linearized line
bundles. Suppose they are homologically equivalent. Then X ss (L) = X ss (L0 ).
97
Proof. This follows immediately from the numerical criterion of stability
and Lemma 2.3.5.
¤
3.1.5 Remark. One should compare this result with Corollary 1.20 from
[MFK]. Under assumption that Hom(G, C∗ ) = {1} it asserts that X s (L) =
X s (L0 ) for any G-linearized line bundles defining the same element in
N S(X) = P ic(X)/P ic(X)0 .
3.1.6 Using the terminology from 2.5.2, we can extend the previous results
and the definitions to any point [ω] from N S G (X)R (not necessary rational). By definition [ω] consists of a cohomology class of some K-invariant
2-form ω of type (1, 1) with non-degenerate imaginary part Im(ω) and a
choice of a moment map defined by the symplectic form Im(ω). The proof
of Theorem 2.3.6 and the discussion of 2.3.8 show that the set X ss ([ω]) is
well defined. It equals the union of orbits which contain in its closure a
point from G · (Φω )−1 (0) where ω is a representative of [ω]. We say that [ω]
is G-effective if X ss ([ω]) 6= ∅. By the convexity of a moment map the set
of G-effective points is closed under addition. It is obviously closed under multiplication by positive scalars. Thus we can define a convex cone
EF G (X) of G-effective points in N S G (X)R . We shall call it the G-effective
cone of X.
98
7.2. The G-ample cone.
Here comes our main definition:
Definition 7.2.1. 3.2.1 Definition. The G-ample cone (for the action of G on
X) is the convex cone in N S G (X)R spanned by the subset N S G (X)+ . It is
denoted by C G (X).
3.2.2 Let X be a complex manifold. The subset of the space H 1,1 (X, R)
formed by the classes of Kähler forms is an open convex cone spanning
the space. It is called the Kähler cone. Its integral points are the classes of
Hodge Kähler forms. By a theorem of Kodaira, each such class is the first
Chern class of an ample line bundle L. The subcone of the Kähler cone
spanned by its integral points is called the ample cone and is denoted by
A1 (X)+ . It is not empty if and only if X is a projective algebraic variety.
1,1
It spans the subspace A1 (X)+
(X, R) formed by the cohomology
R of H
classes of algebraic cycles of codimension 1. The dimension of this subspace is equal to the Picard number of X. The closure of the ample cone
consists of the classes of numerically effective (nef) line bundles [Kl]. Recall that a line bundle is called nef if its restriction to any curve is an effective line bundle. Under the forgetful map N S G (X)R → N S(X)R , the
G-ample cone is contained in the closure of the ample cone. Of course they
are equal if G = {1}. Summing up we conclude that
C G (X) = EF G (X) ∩ α−1 (A1 (X)+
R ),
where α : N S G (X)R → N S(X)R → H 1,1 (X, R) is the composition map.
3.2.3 Remark. There could be no G-effective G-linearized line bundles so
C G (X) could be empty. The simplest example is any homogeneous space
X = G/P , where P is a parabolic subgroup of a reductive group G.
3.2.4 Let us recall some terminology from the theory of convex sets (see
[Ro]). A convex set is a subset S of a real vector space V such that for any
x, y ∈ S and any nonnegative real numbers λ, µ with λ + µ = 1, λx + µy ∈
S. A convex set is a convex cone if additionally for each x ∈ S and any
nonnegative real number λ, λx ∈ S. The minimal dimension of an affine
subspace containing a convex cone S is called the dimension of S and is
denoted by dim(S). The relative interior ri(S) of S is the interior of S in
the sense of the topology of the minimal affine space containing S. The
complementary set r∂(S) := S \ ri(S) is called the relative boundary of S.
A convex cone S is closed if its boundary is empty. A closed convex cone is
called polyhedral if it is equal to the intersection of a finite number of closed
half-spaces. A linear hyperplane is called a supporting hyperplane at a point
x ∈ ∂(S) if S lies in one of the two half-spaces defined by this hyperplane.
This half-space is called the supporting half-space at x. A face of a closed
convex set S is the intersection of S with a supporting hyperplane. Each
99
face is a closed convex set. A function f : V → R ∪ {∞} is called lower
convex if f (x + y) ≤ f (x) + f (y) for any x, y ∈ V . It is called positively
homogeneous if f (λx) = λf (x) for any nonnegative λ. If f is a lower convex
positively homogeneous function, then the set {x ∈ V : f (x) ≤ 0} is a
closed convex cone. For any set A we denote by hAi the smallest closed
convex cone containing A.
Lemma 7.2.2. 3.2.5 Lemma.
(i) The relative interior of any non-empty convex cone of positive dimension is
nonempty;
(ii) A point belongs to the boundary of a convex set S if and only if there exists a
supporting hyperplane at this point;
(iii) Each closed convex cone is equal to the intersection of the supporting halfspaces chosen for each point of its boundary;
(iv) The closure of a convex set is a convex set.
Now we can go back to our convex set C G (X).
Lemma 7.2.3. 3.2.6 Lemma. For each x ∈ X the function P icG (X) → Z, L →
M L (x) factors through N S G (X) and can be uniquely extended to a positively
homogeneous lower convex function M • (x) : N S G (X)R → R.
0
Proof. By Lemma 2.3.5, M L (x) = M L (x) if L is homologically equivalent
to L0 . This shows that we can descend the function L → M L (x) to the
factor group N S G (X). By using 1.4.5 (v), we find
⊗n
M L (x) = nM L (x)
for any nonnegative integer n, and
0
0
M L⊗L (x) = supλ
0
µL⊗L (x, λ)
µL (x, λ) + µL (x, λ)
= supλ (
)
kλk
kλk
0
µL (x, λ)
µL (x, λ)
0
+ supλ
= M L (x) + M L (x).
kλk
kλk
Let us denote by [L] the class of L ∈ P icG (X) in N S G (X). If we put
1
1
M n [L] (x) = M L (x) if n is positive,
n
1
1
−1
M L (x) if n is negative,
M n [L] (x) =
−n
then we will be able to extend the function L → M L (x) to a unique function on
N S G (X)Q = N S G (X) ⊗ Q
satisfying
0
0
M l+l (x) ≤ M l (x) + M l (x),
≤ supλ
100
M αl (x) = αM l (x)
for any l, l0 ∈ N S G (X)Q and any non-negative rational number α.
PNow let
G
us choose a basis e1 , . . . , enPin N S (X)Q and define a norm by k i ai ei k =
supi |ai |. Then for any l = i ai ei we have
X
X
M l (x) ≤
M ai ei (x) ≤
sup{|ai |M ei (x), |ai |M −ei (x)} ≤ Cklk
i
i
for some constant C independent of l. This implies that
l
M klk (x) ≤ C.
l
But M klk (x) is clearly bounded from below by the minimum of the conl
l
klk
µ klk (x,λ)
kλk
tinuous function
:→
defined over a sphere of radius 1 for any
fixed λ. Since a bounded convex function is continuous in N S G (X)Q , the
above argument allows us to extend the function M • (x) to a unique continuous function on N S G (X)R . Obviously it must be lower convex and
positively homogeneous.
¤
Proposition 7.2.4. 3.2.7 Proposition. Assume dim C G (X) = dim N S G (X)R .
For each x ∈ X the restriction of the function M • (x) to C G (X) coincides with
the function [ω] → M ω (x), where ω is any representative of [ω] and M ω (x) is
defined in 2.5.2.
Proof. The two functions coincide on the set of rational points in C G (X).
¤
What can we say about the boundary of C G (X)? It is clear that any
integral point in it is represented either by an ample line G-bundle or by a
nef but non-ample G-bundle L.
Proposition 7.2.5. 3.2.8 Proposition. Let L be an ample G-linearized line bundle
in the boundary of C G (X). Then L ∈ C G (X) and X s (L) = ∅.
Proof. First let us show that X s (L) = ∅. Suppose x ∈ X s (L), then M L (x) <
0 and hence the intersection of the open set {[ω] : M [ω] (x) < 0} with
the open cone α−1 (A1 (X)+
R ) is an open neighborhood of [L] contained in
C G (X). This contradicts the assumption that L is on the boundary.
Now let us show that L ∈ C G (X), i.e., X ss (L) 6= ∅. Suppose X ss (L) =
∅. This means that M L (x) > 0 for all x ∈ X. By Theorem 1.3.9 the set
S of open subsets U of X which can be realized as the set X ss (M ) for
some ample G-linearized line bundle M is finite. Choose a point xU from
each such U . Then X ss (L) = ∅ if and only if M L (xU ) > 0 for each U .
This shows that L is contained in the intersection of the open set {[ω] :
M [ω] (xU ) > 0, U ∈ S} with the open cone α−1 (A1 (X)+
R ). Thus L belongs to
101
the complement of the closure of C G (X) contradicting the assumption on
L.
¤
Corollary 7.2.6. 3.2.9 Corollary. Assume that there exists an ample G-linearized
line bundle L with X s (L) 6= ∅. Then the interior of C G (X) is not empty.
102
7.3. Walls and chambers. .
Definition 7.3.1. 3.3.1 Definition. A subset H of C G (X) is called a wall if
there exists a point x ∈ X(>0) := {x : dimGx > 0} such that H = H(x) :=
{l ∈ C G (X) : M l (x) = 0}. A wall is called proper if it is not equal to the
whole cone C G (X). A connected component of the complement of the
union of walls in C G (X) is called a chamber.
It is clear that all walls are proper if there exists an ample G-linearized
line bundle with
X s (L) = X ss (L).
Theorem 7.3.2. 3.3.2 Theorem. Let l, l0 be two points from C G (X).
(i) l belongs to some wall if and only if X sss (l) 6= ∅;
(ii) l and l0 belong to the same chamber if and only if X s (l) = X ss (l) = X ss (l0 ) =
X s (l0 );
(iii) each chamber is a convex cone.
Proof. (i) If l belongs to some wall then there exists a point x ∈ X such that
M l (x) = 0. By 2.5.2, x ∈ X sss (l). Conversely, if x ∈ X sss (l) then the closure
of G · x contains a closed orbit G · y in X sss (l) with stabilizer Gy of positive
dimension. Thus l lies on the wall H(y).
(ii) If l belongs to a chamber C, then X s (l) = {x ∈ X : M L (l) < 0}. The
function M • (x) does not change sign in the interior of C. Because if it does,
0
we can find a point l0 in C such that M l (x) = 0. Arguing as in the proof
of (i) we find that l0 belongs to a wall. This shows that the set X s (l) does
not depend on l. We shall denote it by X s (C). Let us prove the converse.
Assume X s (C) = X s (C 0 ) for two chambers C and C 0 . We have C, C 0 ⊂
∩x∈X s (C) {M • (x) < 0}. Pick l in ∩x∈X s (C) {M • (x) < 0}. Then X s (C) ⊂ X s (l),
hence we get X s (C)/G ⊂ X s (l)/G. By (i), the first quotient is compact.
Therefore the second quotient is compact and X s (l) = X ss (l). This shows
that l does not belong to the union of walls. Since ∩x∈X s (C) {M • (x) < 0}
is convex, and hence connected, it must coincide with C. This shows that
C = C 0 and the set X s (C) determines the chamber C uniquely.
(iii) This obviously follows from the fact that the function M • (x) does not
take the value zero in a chamber and is positively homogeneous lower
convex.
¤
Theorem 7.3.3. 3.3.3 Theorem. There are only finitely many walls.
Proof. For any wall H let X(H) = {x ∈ X(>0) : H = H(x)} where X(>0) is
the set of points in X with positive dimensional stabilizer. Obviously
X(H) = ∩l∈H (X ss (l)) ∩ X(>0) ).
103
By Theorem 2.4.8, we can find a finite set of points l1 , . . . , lN in C G (X)
such that for any l ∈ C G (X), the set X ss (l) equals one of the sets X ss (li ).
Using the above description of X(H) we obtain that there are only finitely
many subsets of X which are equal to the subset X(H) for some wall H.
However, two walls H, H 0 are equal if and only if X(H) = X(H 0 ) if and
only if X(H) ∩ X(H 0 ) 6= ∅. This proves the assertion.
¤
Proposition 7.3.4. 3.3.4 Proposition. Each proper wall is equal to a countable
union of convex closed cones of dimension strictly less than the dimension of
C G (X).
Proof. A wall is a set of the form {l ∈ C G (X)R : M l (x) = 0}. The set
{l ∈ C G (X)R : M l (x) ≤ 0} is a convex cone. The function M • (x) is defined as the supremum of linear functions µl (x, λ)/||λ||. Those of them
which are not identically zero are obviously the supporting hyperplanes
of the cone. It follows from the properties of these functions stated in
1.1.1 that there are only countably many different functions among them
(see [MFK],p.59). It remains to use that the boundary of a convex cone
C is equal to the union of intersections of C with its supporting hyperplanes.
¤
Since a wall is the union of convex cones we can speak about its dimension (being the maximum of the dimensions of the cones).
Proposition 7.3.5. 3.3.5 Proposition. Assume all walls are proper. Then X s (l) 6=
∅ for any l in the interior of C G (X).
Proof. Assume l is contained in an open subset U of C G (X). By Theorem
3.3.3, the union of walls is a proper closed subset of C G (X). If l does not
belong to any wall, the assertion is obvious. If it does, we choose a line
through l which contains two points l1 , l2 ∈ U which do not lie in the
union of walls. Then X s (l1 ) ∩ X s (l2 ) 6= ∅ so that M l1 (x) < 0 and M l2 (x) < 0
for some x ∈ X. By convexity, M l (x) < 0, hence x ∈ X s (l).
¤
Together with Proposition 3.2.8 this gives the following
Corollary 7.3.6. 3.3.6 Corollary. Assume all walls are proper. The intersection of
the boundary of C G (X) with the ample cone is the set {l ∈ C G (X) : X s (l) = ∅}.
Lemma 7.3.7. 3.3.7 Lemma. Let H be a wall. There exists a point x ∈ X such
that H = H(x) but H 6= H(y) for any y ∈ G · x \ G · x.
Proof. Take any point x ∈ X such that H = H(x). Let y ∈ G · x \ G · x.
Assume H = H(y). Since dim G · y < dim G · x, we can replace x by y and
continue this process until we find the needed point.
¤
104
Definition 7.3.8. 3.3.8 Definition. A point x ∈ X is called a pivotal point for
a wall H if H = H(x) and H 6= H(y) for any y ∈ G · x \ G · x.
Lemma 7.3.9. 3.3.9 Lemma. Let Φ : X → Lie(K)∗ be a moment map associated
to some symplectic structure on X. For any x ∈ Φ−1 (0) the connected component
of Gx is a reductive algebraic group. In particular, if G·x is a closed orbit in X ss (l)
for some l ∈ C G (X), then the connected component of Gx is a reductive algebraic
group.
Proof. This is Lemma 2.5 from [Ki1]. In the case when the moment map
is defined by an ample line bundle L the proof is as follows. By Theorem
2.2.1, G · x is closed in X ss (L). Let π : X ss (L) → X ss (L)//G be the quotient
projection. Its fibers are affine, and hence G · x is a closed subset of an
affine variety. Thus G · x = G/Gx is an affine variety. Now the assertion
follows from [B-B1].
¤
Proposition 7.3.10. 3.3.10 Proposition. Let x be a pivotal point for a wall H.
Then the connected component of Gx is a reductive subgroup of G.
Proof. Let l ∈ H ∩ C G (X) but l ∈
/ H(y) for any y ∈ G · x \ G · x. Since the
number of walls is finite, we can always find such l. Now, by the choice of
l, one sees that G·x is a closed orbit in X ss (l) and we can apply the previous
lemma to obtain that the connected component of Gx is reductive.
¤
Definition 7.3.11. 3.3.11 Definition. Let S be the set of all possible moment
map stratifications of X defined by points l ∈ C G (X). There is a natural
partial order in S by refinement. We say that s refines s0 if any stratum of
s0 is a union of strata of s. In this case, we sometimes write s < s0 if s 6= s0 .
Let C be a subset of C G (X). For any s ∈ S, let S
Cs be the set of elements
l ∈ C which define the stratification s. Then C = s∈S Cs .
The next proposition implies that s refines any element in the closure of
Cs .
Proposition 7.3.12. 3.3.12 Proposition. Let ln ∈ C G (X) be a sequence of points
in C G (X) that induce the same stratification s ∈ S. Assume that ln → l ∈
C G (X). Then s refines the stratification s0 induced by l.
min
, it suffices to show
Proof. Let Sβ(ln ) be a stratum in s. Since Sβ(ln ) = GYβ(l
n)
min
0
that Yβ(ln ) is contained in a stratum of s . Let λ(ln ) be the one-parameter
subgroup generated by β(ln ). Then, by Theorem 2.4.7, we can also write
min
Yβ(l
as Sdn ,λ(ln ) , where dn = ||β(ln )||. By passing to a subsequence, we
n)
may assume that dn → d, β(ln ) → β(l). Let λ(l) be the one-parameter
min
, we have M ln (x) =
subgroup generated by β(l). For any point x ∈ Yβ(l
n)
µ̄ln (x, λ(ln )) = dn (see Theorem 2.1.9 (iii)). By taking limit, we find that
105
min
M l (x) = µ̄l (x, λ(l)) = d. If d = 0, we obtain that Yβ(l
⊂ X ss (l). If d 6= 0,
n)
min
we obtain that Yβ(l
⊂ Sd,<λ(l)> . This completes the proof.
¤
n)
The following will play the key role in the sequel.
Lemma 7.3.13. 3.3.13 Lemma. Let l0 be in the closure of a chamber C. Then
there is a sequence of points ln in C which induce the same stratification s and
ln → l0 . In particular, X ss (l0 ) is a union of the strata of s.
Proof. Take a basis U1 ⊃ U2 ⊃ · · · ⊃ Un ⊃ · · · of open subsets containing
l0 . We have that Un ∩ C 6= ∅ and is open in C. Write S = {s1 , · · · , sm }.
If s1 occurs as the induced stratification for some point ln in Un ∩ C for
every n, then we are done. Suppose it does not. Then there is n1 such that
(Un1 ∩ C)s1 = ∅. Consider s2 . If s2 occurs as the induced stratification for
some point ln in Un ∩ C for every n ≥ n1 , then we are done. Suppose it
does not. Then there is n2 ≥ n1 such that (Un2 ∩ C)s2 = ∅. The next step
is to consider s3 , and so on. Since the set S = {s1 , · · · sm } is finite and
U1 ∩ C ⊃ U2 ∩ C ⊃ · · · ⊃ Un ∩ C ⊃ · · · is infinite, there must be si ∈ S such
that si occurs as the induced stratification for some point ln in Un ∩ C for
every n ≥ ni−1 .
Then {ln } (n ≥ ni−1 ), is the desired sequence.
¤
3.3.14 For any x ∈ X and L ∈ P icG (X) let ρx (L) : Gx → GL(Lx ) ∼
= C∗ be
the isotropy representation. For any λ ∈ X∗ (Gx ) the composition
hλ, ρLx i = λ ◦ ρLx : C∗ → C∗
L
is equal to the map t → tµ (x,λ) . Obviously ρx (L) is trivial if L is Ghomologically trivial. Thus we can define a homomorphism
ρx : N S G (X) → X (Gx ), L → ρx (L).
By linearity we can extend ρx to a linear map
ρx : N S G (X)R → X (Gx ) ⊗ R.
For any l ∈ N S G (X)R and λ ∈ X∗ (Gx ) we have, by continuity,
l
hλ, ρlx i(t) = tµ (x,λ) .
We denote by Ex the kernel of this homomorphism and by Ex the vector
subspace of N S G (X)R generated by Ex .
Proposition 7.3.14. 3.3.15 Proposition. Let x be a pivotal point of a wall H.
Assume that all walls are proper. Then Ex 6= N S G (X)R .
Proof. Let l be chosen as in the proof of Proposition 3.3.10. Since all walls
are proper, we can find an open neighborhood U of l in N S G (X)R which
106
contains l0 ∈ C G (X) which does not belong to any wall, and the stratification induced by l0 refines the stratification induced by l (see Lemma
3.3.13). We can also find a continuous path in U which starts at l0 , ends at
l and does not cross any walls. Since M • (x) is a continuous function, we
0
0
have M l (y) < 0 implies M l (y) ≤ 0, and M l (y) < 0 implies M l (y) < 0.
This shows that
X s (l0 ) ⊂ X ss (l), X s (l) ⊂ X s (l0 ).
In particular, X us (l) ⊂ X us (l0 ). Let us now use the moment map stratification for X with respect to l and l0 . We can write
X = X ss (l) ∪ X us (l) = X s (l0 ) ∪ Sβ1 ∪ . . . ∪ Sβk ∪ X us (l).
Here, we use the fact that the strata of X us (l) coincide with the union of
some strata of X with respect to l0 . Now the point x must belong to some
strata Sβi since it is semi-stable with respect to l but unstable with respect
to l0 . Let us denote βi by λ. We claim that λ(C∗ ) is contained in Gx . Indeed,
if not, y = limt→0 λ(t) · x does not belong to the orbit G · x but is contained
in Sλ ⊂ X ss (l). But this contradicts the choice of l. This proves our claim.
0
0
Now λ ∈ Λl (x) hence M l (x) =
l0 ∈
/ Ex .
0
µl (x,λ)
kλk
0
> 0. This gives µl (x, λ) > 0 hence
¤
Corollary 7.3.15. 3.3.16 Corollary. Assume all walls are proper. Let H be a wall
and x be its pivotal point. Then
(i) Gx is a reductive group with non-trivial radical R(Gx );
(ii) H ⊂ Ex ;
(iii) if codimH = 1 then H equals the closure of its set of rational points.
Proof. The first assertion obviously follows from the fact that Ex 6= N S G (X)
implies X (Gx ) 6= {1}.
Let us prove (ii). Take any l ∈ H(x). Since x ∈ X ss (L), for any oneparameter subgroup λ in Gx we have µl (x, λ) = 0. In fact, otherwise
µl (x, λ) or µl (x, λ−1 ) is positive, and then x ∈ X us (L). This shows that
the composition of λ : C∗ → Gx with ρx (L) : Gx → C∗ is trivial. Hence
l ∈ Ex .
To prove (iii) we use that, by (ii), H is contained in Ex . Since Ex is a
proper subspace, H spans it. This implies that the relative interior ri(H)
of H contains an open subset of Ex . Since Ex can be defined by a linear equation over Q this open subset contains a dense subset of rational
points.
¤
Proposition 7.3.16. 3.3.17 Proposition. Assume that all walls are proper. Then
any wall is contained in a codimension 1 wall.
107
Proof. Since the union of walls is a closed subset of C G (X), each chamber
is an open subset of C G (X). Suppose there is a wall H of codimension
≥ 2 which is not contained in any codimension 1 wall. Let l be a point
in the relative interior of H. Then there exists an open subset U of C G (X)
containing l such that U \H∩U is contained in the complement of the union
of walls. Since H is of codimension ≥ 2, the set U \ H ∩ U is connected.
Hence there exists a chamber C containing this set. Let l1 , l2 ∈ C such that
l is on the line joining l1 with l2 . As in the proof of Proposition 3.3.5,
X s (l1 ) ∩ X s (l2 ) ⊂ X s (l).
By Theorem 3.3.2 (ii), we see that the left hand side is equal to X ss (l1 ).
Since the quotient X ss (l1 )//G = X s (l1 )/G is compact, we obtain X s (l)/G
is compact, hence X ss (l) = X s (l). But this contradicts the assumption that
l belongs to a wall.
¤
Theorem 7.3.17. 3.3.18 Theorem. Assume that all walls are proper. Then the
intersection of the boundary of a chamber with interior of C G (X) is contained in
the union of finitely many rational supporting hyperplanes.
Proof. Let l be an interior point of C G (X) which belongs to the boundary
of a chamber C. Let H1 , . . . , HN be codimension 1 walls containing l. We
claim that for at least one of these walls H = Hi the closure C̄ of C contains a point in the relative interior of Hi . In fact, otherwise l is contained
in an open neighborhood U such that the intersection of U with the union
of chambers is connected. Since U is not contained in C this contradicts
the definition of chambers. By Proposition 3.3.15 and Corollary 3.3.16, H
spans a rational hyperplane Ex , where x is a pivotal point of H. This hyperplane is the needed supporting hyperplane for the closure C̄ of C at
the point l. To verify this it suffices to show that C ∩ Ex = ∅. Suppose l0
is in the intersection. Choose a point l00 in the relative interior of H which
belongs to the boundary of C. Then the segment of the line joining the
points l0 with l00 is contained in Ex which is spanned by H. On the other
hand it contains only one point from H, the end point l00 . Since l00 ∈ ri(H),
this is obviously impossible.
¤
3.3.19 Remark. Theorem 3.3.18 shows that each chamber is a rational convex polyhedron away from the boundary of C G (X). There is no reason
to expect that the boundary of C G (X) is polyhedral. This is not true even
when G is trivial. However some assumptions on X, for example X is a
Fano variety (the dual of the canonical line bundle is ample) may imply
that the closure of C G (X) is a convex polyhedral cone.
3.3.20 Example. Let G be an n-dimensional torus (C∗ )n , acting on X =
P(V ) via a linear representation ρ : G → GL(V ). Then N S G (X) ∼
= P ic(X)×
X (G) ∼
= Zn+1 . The splitting is achieved by fixing the G-linearization on
108
OX (1) defined by the linear representation ρ. In addition, N S G (X)+ =
Z>0 × N S G (X)1 , where N S G (X)1 is the group of G-linearizations on the
line bundle OX (1) identified with X (G). Thus L0 = (OX (1), ρ) corresponds
to the zero in X (G). By 1.1.5, X ss (L0 ) 6= ∅ (resp.X s (L0 ) 6= ∅) if and only
if 0 ∈ St(V ) (resp. 0 ∈ int(St(V )) (see 2.1.8 for the notation St(V )).
If Lχ is defined by twisting the linearization of L0 by a character χ, we
obtain that X ss (Lχ ) 6= ∅ (resp.X s (Lχ ) 6= ∅) if and only if χ−1 ∈ St(V )
(resp.χ−1 ∈ int(St(V ))). In particular, we get that C G (X) is equal to the
cone over the convex hull St(V ) of St(V ). Comparing this with 2.1.8, we
obtain that C G (X) is equal to the cone over the image of the moment map
for the Fubini-Study symplectic form on X.
For any x ∈ X the image of the moment map of the orbit G · x is equal
to the subpolytope P (X) of St(V ) which is equal to the convex hull of the
state set st(x) of x. The codimension of P (X) is equal to dim Gx > 0. A
wall H(x) is the cone over P (x) with dim Gx > 0. The union of walls is
equal to the cone over the set of critical points of the moment map.
3.3.21 Example. Let G = SL(n + 1) act diagonally on X = (Pn )m . We
have P ic(X) ∼
= P icG (X) ∼
= Z m . Each line bundle L over X is isomorphic
to the line bundle Lk = p∗1 (OPn (k1 )) ⊗ . . . ⊗ p∗m (OPn (km )) for some k =
(k1 , . . . , km ) ∈ Zm . Here pi : X → Pn denotes the projection map to the i-th
factor. It is easy to see that Lk is nef (resp. ample) if and only if all ki are
nonnegative (resp. positive) integers. Let P = (p1 , . . . , pm ) ∈ X. Using the
numerical criterion of stability, one easily gets (see [Do]) that P ∈ X ss (Lk )
if and only if for any proper linear subspace W of Pn
m
dim W + 1 X
ki ≤
(
ki ).
n
+
1
i=1
i,p ∈W
X
i
The strict inequality characterizes stable points. This easily implies that
ss
X (Lk )) 6= ∅ ⇔ (n + 1)maxi {ki } ≤
m
X
ki .
i=1
Let ∆n,m = {x = (x1 , . . . , xm ) ∈ Rm :
m
P
xi = n + 1, 0 ≤ xi ≤ 1, i =
i=1
1, . . . , m}. This is the so-called (m − 1)-dimensional hypersimplex of type n.
Consider the cone over ∆n,m in Rm+1
C∆n,m = {(x, λ) ∈ Rm × R+ : x ∈ λ∆n,m }.
We have the injective map
P icG (X) → Rm+1 , Lk 7→ (k1 , . . . , km , (n + 1)−1
m
X
i=1
109
ki ),
which allows us to identify P icG (X) with a subset of Rm+1 . We have
P icG (X) ∩ C∆n,m = { L ∈ P icG (X) : L is nef, X ss (L) 6= ∅}.
Thus the G-ample cone C G (X) coincides with the interior of C∆n,m .
Observe that P ∈ X sss (Lk ) if and only if there exists a subspace W of
dimension d, 0 ≤ d ≤ n − 1 such that
m
X
X
(n + 1)
ki = (dim W + 1)
ki .
i=1
i,pi ∈W
This is equivalent to the condition that Lk belongs to the hyperplane
X
HI,d := {(x1 , . . . , xm , λ) ∈ Rm :
xi = λd},
i∈I
where I is a non-empty subsetSof {1, . . . , m}. Thus a chamber is a connected component of C∆n,m \ I,d HI,d . A wall is defined by intersection
of the interior of C∆n,m with a subspace
X
xi = λdj , j = 1, . . . , s},
HI1 ,...,Is ,d1 ,...,ds := {(x1 , . . . , xm , λ) ∈ Rm :
`
i∈Ij
`
where {1, . . . , m} = I1 . . . Is , d1 + . . . + ds = n + 1 − s, s ≥ 2.
Note that the set ∆n,m is the image of the moment map for the natuV
ral action of the torus T = (C∗ )m on P( n+1 Cm ). Comparing with Example 3.3.20, we obtain that the closure of C SL(n+1) ((Pn )m ) is equal to
V
C T (P( n+1 Cm )). There is a reason for this. For any P = (p1 , · · · , pm ) ∈
(Pn )m one can consider a matrix A of size (n+1)×m whose i-th column is a
vector in C n+1 representing the point pi . Let E(A) be the point of the GrassV
mann variety G(n + 1, m) ⊂ P( n+1 Cm ) defined by the matrix A. The
different choice of projective coordinates of the points pi replaces E(A) by
the point t · E(A) for some t ∈ T . In this way we obtain a bijection between
SL(n + 1)-orbits of points (p1 , . . . , pm ) ∈ (Pn )m with hp1 , . . . , pm i = Pn and
orbits of T on G(n + 1, m). This is called the Gelfand-MacPherson correV
spondence. The T -ample cone C T (G(n + 1, m)) equals C T (P( n+1 Cm ) and
hence coincides with the closure of the SL(n + 1)-ample cone of (Pn )m .
The Gelfand-MacPherson correspondence defines a natural isomorphism
between the two GIT-quotients corresponding to the same point in C∆n,m .
110
7.4. GIT-equivalence classes. .
Definition 7.4.1. 3.4.1 Definition. Two elements l and l0 in C G (X) are called
GIT-equivalent (resp. weakly GIT-equivalent) if X ss (l) = X ss (l0 ) (resp.
X s (l) = X s (l0 )).
Let E ⊂ C G (X) be a GIT-equivalence class. We denote by X ss (E) the
subset of X equal to X ss (l) for any l ∈ E. Clearly for any l ∈ E the subset X s (l) is equal to the subset of points in X ss (E) whose orbit is closed
in X ss (E) and the stabilizer is finite. This shows that this set is independent of a choice of l, so we can denote it by X s (E). In particular, GITequivalence implies weak GIT-equivalence. Similarly we introduce the
subset X ss (E)(>0) of points in X ss (E) whose stabilizers are of positive dimension. It is clear that
E ∩ H(x) 6= ∅ ⇔ E ⊂ H(x) ⇔ x ∈ X ss (E)(>0)
where x is a pivotal point for the wall H(x). Examples of GIT-equivalence
classes are chambers (Theorem 3.3.2). For these equivalence classes X ss (E) =
X s (E). Chambers are the only GIT-equivalence classes which coincide
with weak GIT-equivalence classes.
Theorem 7.4.2. 3.4.2 Theorem. Let H be a wall and H ◦ be the subset of points
from H which do not lie in any wall not containing H. Assume H ◦ 6= ∅. Then
each connected component of H ◦ is contained in a GIT -equivalence class. Conversely, for each equivalence class E which is not a chamber there exists a wall H
such that E is the union of some connected components of H ◦ . In particular, a
subset is a connected component of a GIT -equivalence class if and only if it is a
chamber or a connected component of H ◦ for some wall H.
Proof. Obviously, l, l0 ∈ H ◦ if and only if X sss (l) = X sss (l0 ). For any x ∈ X
the function M • (x) either vanishes identically on H ◦ or does not take the
value zero at any point. This implies that this function does not change its
sign at any connected component F of H ◦ . Thus for any l, l0 ∈ F , X s (l) =
X s (l0 ). Together with X sss (l) = X sss (l0 ) we obtain that F is contained in a
GIT-equivalence class. By the previous discussion, any equivalence class
E not equal to a chamber is contained in H ◦ for some H. Hence it must be
equal to the union of connected components of H ◦ .
¤
Lemma 7.4.3. 3.4.3 Lemma. Let l and l0 be two points in C G (X). If X ss (l) ⊂
X ss (l0 ), then X s (l0 ) ⊂ X s (l). Consequently, if X ss (l) ⊂ X ss (l0 ) and X s (l) ⊂
X s (l0 ), then l and l0 are weakly equivalent.
Proof. If X s (l0 ) = ∅, then there is nothing to prove. Assume that X s (l0 ) 6= ∅.
We claim that the inclusion X ss (l) ⊂ X ss (l0 ) implies that X s (l) 6= ∅. Otherwise, we would have X ss (l) = X sss (l) ⊂ X sss (l0 ) because a semistable
point is strictly semistable if and only if its orbit closure contains a semistable
111
point with isotropy of positive dimension. The inclusion X ss (l) ⊂ X sss (l0 )
is impossible because X ss (l) is Zariski open in X, whereas X sss (l0 ) is not
(noting that X s (l0 ) 6= ∅).
Now the inclusion X ss (l) ⊂ X ss (l0 ) obviously induces a morphism f :
X ss (l)//G → X ss (l0 )//G. This map sends X sss (l)//G to X sss (l0 )//G. Since
f is birational and hence surjective, for any point in X ss (l0 )//G representing the orbit of a stable point, the fiber of f at this point must be a point
representing the orbit of a stable point in X s (l). This is not possible unless X s (l0 ) ⊂ X s (l). In particular, if in addition X s (l) ⊂ X s (l0 ), the above
implies that X s (l) = X s (l0 ).
¤
Theorem 7.4.4. 3.4.4 Theorem. Let E be a GIT-equivalent class which is contained in some wall H. Then its convex hull CH(E) is contained in a weak GITequivalent class. CH(E) \ E is contained in H \ H ◦ .
Proof. Let l, l0 ∈ E be joined by a segment S of a straight line and l00 be a
point on this segment. By the lower convexity of the functions M • (x) we
have X s (l) ∩ X s (l0 ) ⊂ X s (l00 ) and X ss (l) ∩ X ss (l0 ) ⊂ X ss (l00 ). Since X ss (l) =
X ss (l0 ) and hence X s (l) = X s (l0 ), we obtain X s (l) ⊂ X s (l00 ), X ss (l) ⊂
X ss (l00 ). By the previous lemma, l and l00 are weakly GIT-equivalent. Clearly
CH(E) ⊂ H. Assume that l00 does not lie on any wall not containing H.
Then the proof of the previous theorem shows that X ss (l) = X ss (l00 ), hence
l, l00 ∈ E. This shows that all points on S lying on H ◦ are in the equivalence
class E. Hence CH(E) \ E ⊂ H \ H ◦ .
¤
Definition 7.4.5. 3.4.5 Definition. A non-empty subset of C G (X) is called
a cell if it is a chamber or a connected component of H ◦ for some wall H.
By Theorem 3.4.2, one can give an equivalent definition by defining a
cell to be a connected component of a GIT equivalence class.
Definition 7.4.6. 3.4.6 Definition. A pair of chambers (C, C 0 ) are called
relevant to a cell F if F ⊂ C ∩ C 0 and there is a straight path l : [−1, 1] →
C G (X) such that l([−1, 0)) ⊂ C, l(0) ∈ F and l((0, 1]) ⊂ C 0 .
Proposition 7.4.7. 3.4.7 Proposition. Let (C, C 0 ) be a pair of chambers relevant
to a cell F . Then,
X s (F ) = X s (C) ∩ X s (C 0 ), X ss (F ) ⊃ X s (C) ∪ X s (C 0 ).
Proof. The fact that X ss (F ) ⊃ X s (C)∪X s (C 0 ) and X s (F ) ⊃ X s (C)∩X s (C 0 )
follows from the lower convexity of the function M • (x). The fact that
X s (F ) ⊂ X s (C) ∩ X s (C 0 ) follows from the continuity of M • (x).
¤
112
7.5. Abundant actions.
Definition 7.5.1. 3.5.1 Definition. We say that P icG (X) ( or the action) is
abundant if for any pivotal point x of any wall the isotropy homomorphism
ρx : P icG (X) → X (Gx ) has finite cokernel.
The abundance helps to control the codimensions of walls.
Proposition 7.5.2. 3.5.2 Proposition. Assume that P icG (X) is abundant. Then
for any pivotal point x of a codimension k wall H the radical of the stabilizer group
Gx is of dimension k or less.
Proof. This follows immediately from Proposition 3.3.16.
¤
Theorem 7.5.3. 3.5.3 Theorem. Let B be a Borel subgroup of G and let G act on
X × G/B diagonally. Then P icG (X × G/B) is abundant. In particular, when G
is a torus, then P icG (X) is abundant.
Proof. Notice first that P icG (G/B) ∼
= X (T ), where T is a maximal torus
contained in B (see [KKV], p.65). We want to show that for any point
(x, g[B]) with reductive stabilizer the isotropy representation homomorphism
ρ(x,g[B]) : P icG (X × G/B) → X (G(x,g[B]) )
is surjective. Obviously G(x,g[B]) = Gx ∩ Gg[B] . By conjugation, we may
assume that R(x,g[B]) ⊂ Rx ⊂ T , where Rx denotes the reductive radical of
the stabilizer subgroup Gx . Pick any character χ ∈ X (G(x,g[B]) ). Extend it
to a character χ̃ ∈ X (T ). Let Lχ̃ ∈ P icG (G/B) be the line bundle defined by
the character χ̃, L0χ̃ be the induced linearized line bundle in P ic(G/B), and
let L0 be its lift to P icG (X × G/B). Then ρ(x,g[B]) (L0 ) = χ̃|R(x,g[B]) = χ|R(x,g[B]) .
Thus ρ(x,g[B]) is surjective. Hence P icG (X × G/B) is abundant.
The last statement follows immediately because G = B when G is a
torus.
¤
113
8. VARIATION OF GIT
QUOTIENTS
8.1. Faithful walls.
Definition 8.1.1. 4.1.1 Definition. A codimension 1 wall H is called faithful
if for any pivotal point x with H(x) = H the reductive radical of Gx is onedimensional, truly faithful if the stabilizer Gx is one-dimensional, perfect if
it is truly faithful and in addition all such stabilizers are conjugate to each
other. A cell E is called faithful (truly faithful, perfect) if it is contained in
a faithful (truly faithful, perfect) wall.
4.1.2 Example Let G be a reductive algebraic group of rank 1 (e.g., G =
SL(2)) acting on a projective variety X. Assume that all walls are proper.
Then each wall is perfect. In fact by Proposition 3.3.15 the stabilizer of its
pivotal point has non-trivial radical. Since G is of rank 1, all proper reductive subgroups must be tori, all tori are one-dimensional and conjugate to
each other.
Proposition 8.1.2. 4.1.3 Proposition. Let G be a reductive algebraic group acting
on a nonsingular projective variety X. Assume that P icG (X) is abundant. Then
all codimension 1 walls are faithful.
Proof. Follows from Corollary 3.3.16.
¤
Corollary 8.1.3. 4.1.4 Corollary. Let G be a torus. Then all codimension 1 walls
are truly faithful.
Proof. Follows from 3.5.3.
¤
Corollary 8.1.4. 4.1.5 Corollary. Let G act on X × G/B. Then all codimension
1 walls are truly faithful.
Proof. Let (x, g[B]) ∈ X ×G/B. Then G(x,g[B]) = Gx ∩Gg[B] ⊂ Gg[B] = gBg −1 .
Now if (x, g[B]) is a pivotal point for a codimension 1 wall, then G(x,g[B]) is
reductive. This implies that G(x,g[B]) is a torus. Now the corollary follows
from Theorem 3.5.3.
¤
Recall that GIT quotients of X × G/B by G can be identified with symplectic reductions of X by K. This corollary will assure that our main theorem on variation of quotients apply to symplectic reductions for general
coadjoint orbits — and this is exactly what motivates this paper.
114
8.2. GIT wall-crossing flips.
In this section we assume that X is nonsingular and all walls in C G (X)
are proper.
Lemma 8.2.1. 4.2.1 Lemma. Let F be a cell contained in the closure of another
cell E. Then
(i) X ss (E) ⊂ X ss (F );
(ii) X s (F ) ⊂ X s (E);
(iii) the inclusion X ss (E) ⊂ X ss (F ) induces a morphism X ss (E)//G ⊂ X ss (F )//G
which is an isomorphism over X s (F )/G.
Proof. (i) Let l ∈ E and x ∈ X ss (l) = X ss (E). Then M l (x) ≤ 0 so by
0
continuity M l (x) ≤ 0 for any l0 ∈ F . This proves that X ss (E) ⊂ X ss (l0 ) =
X ss (F ).
0
(ii) In the previous notation we have M l (x) < 0 for any x ∈ X s (F ). Thus,
by continuity, M l (x) < 0 for any l ∈ E and hence x ∈ X s (E).
(iii) Follows from (i) and (ii).
¤
4.2.2 Let (C + , C − ) is a pair of chambers relevant to a cell F . Let l0 be a point
in F . Lemma 3.3.13 implies that we can choose l+ ∈ C + and l− ∈ C − such
that their induced stratifications (cf. 1.3 and 2.5) of X can be arranged as
follows
X = X ss (l0 ) ∪ X us (l0 ),
+
+
−
−
X = X s (l+ ) ∪ Sαl 1 ∪ · · · ∪ Sαl p ∪ X us (l0 ),
and
X = X s (l− ) ∪ Sβl 1 ∪ · · · ∪ Sβl q ∪ X us (l0 ).
is conTo simplify the notation we shall assume that each Zαmin
or Zβmin
i
j
nected.
Let
f + : X s (C + )/G → X ss (F )//G, f − : X s (C − )/G → X ss (F )//G
be the morphisms defined in Lemma 4.2.1. They are birational morphisms
of projective varieties which are isomorphisms over the subset X s (F )/G
of X ss (F )//G. The goal of this section is to describe the fibres of the morphisms f + and f − .
Lemma 8.2.2. 4.2.3 Lemma. Keep the previous notation and assume that the cell
F is truely faithful; then we have
(i) p = q;
(ii) Under a suitable arrangement,
Zαmin
= Zβmin
, 1≤i≤q
i
i
115
(iii) αi = −ci βi ( 1 ≤ i ≤ q) for some positive number ci .
−
+
(iv) Sβl j ∩ Sαl i = ∅ if j 6= i.
Proof. First notice that all G-orbits of points from Zαmin
are closed in X ss (l0 ).
i
ss
In fact, otherwise X (l0 ) would contain points with stabilizer of dimension > 1 contradicting the assumption that F is a truly faithful cell. In
addition, notice that αi form the set of one-parameter subgroups (without
parametrizations) of T that have nonempty fixed point set on X sss (l0 )c ,
where X sss (l0 )c is the union of closed orbits in X sss (l0 ). Similar statements
are also true for βj and Zβmin
. This shows that under a suitable arrangej
ment, we may assume p = q and Zαmin
= Zβmin
. This proves (i) and (ii).
i
i
We have already seen that αi and βi generate the same subgroup of G,
i.e., they difer by a constant multiple. Now by the constructions of the
−
+
is the
, where Yαmin
and Sβl i = G · Yβmin
stata, we have Sαl i = G · Yαmin
i
i
i
min
preimage over Zαi under the Bialynicki-Birula contraction determined
under the
= Zαmin
is the preimage over Zβmin
by αi and similarly Yβmin
i
i
i
Bialynicki-Birula contraction determined by βi . If αi and βi differ by a
+
−
positive constant multiple, we would have that Sαl i and Sβl i coincide. Let
+
−
us show that this is impossible. So, assume that Sαl i and Sβl i coincide.
Then pick a point z ∈ Zαmin
. Since the map f − : X s (l− )//G → X s (l0 )//G
i
is surjective, there is a point x ∈ X s (l− ) such that G · z ⊂ G · x. Thus
x ∈ X sss (l0 ). Since x ∈ X s (l− ) ∩ X sss (l0 ), we have that x ∈
/ X s (l+ ) because
X s (l+ ) ∩ X s (l− ) = X s (l0 ) (Proposition 3.4.7). Taking into account the strat+
+
ification X = X s (l+ ) ∪ Sβl 1 ∪ · · · ∪ Sβl q ∪ X us (l0 ), one sees that there must be
+
−
−
j 6= i such that x ∈ Sβl j because we assume that Sαl i = Sβl i . This shows that
such that G · z 0 ⊂ G · x. This contradicts
= Zαmin
there is a point z 0 ∈ Zβmin
j
j
the fact that G · z and G · z 0 are two distinct closed orbits in X ss (l0 ). Therefore αi and βi differ by a negative constant multiple. This proves (ii) and
(iii).
−
+
It remains to show (iv). Assume that there was a point x ∈ Sβl j ∩ Sαl i .
Then there must be a point z ∈ Zβmin
and a point z 0 ∈ Zαmin
such that
i
j
0
ss
G · x ⊃ G · z and G · x ⊃ G · z . Because x ∈ X (l0 ), the above would
imply that G · z and G · z 0 are mapped to the same point in the quotient
X ss (l0 )//G. But this can not happen because G · z and G · z 0 are different
closed orbits in X ss (l0 ).
¤
For convenience, we may assume that αi = −βi . Let λi be the oneparameter subgroup generated by βi and λ−1
be the one-parameter subi
group generated by −βi . Since F is a truly faithful cell, we obtain that
Gz = λi (C∗ ) if z ∈ Zβmin
.
i
116
Lemma 8.2.3. 4.2.4 Lemma. Keep the assumption and the notation from the
previous lemma. Let β ∈ {β1 , . . . , βp }. Then
Yβmin \ U (λ)Zβmin ⊂ X s (l− )
and
min
min
Y−β
\ U (λ−1 )Z−β
⊂ X s (l+ ).
min
min
Proof. Observe first that Zβmin = Z−β
and Yβmin ∩ Y−β
= Zβmin by the
definition.
S
Now since Yβmin ⊂ X = X s (l− ) ∪ i S−βi ∪ X us (l0 ), Sβ ∩ X us (l0 ) = ∅, and
Sβ ∩ S−βi = ∅ if β 6= βi , we have that Yβmin ⊂ X s (l− ) ∪ S−β . Let y ∈ Yβmin .
Then either y ∈ X s (l− ) or y ∈ S−β . Assume that y ∈ S−β . Then there is a
min
g ∈ G and y1 ∈ Y−β
such that y = g · y1 . Because G = U (λ)P (λ−1 ), we can
write g = u · p with u ∈ U (λ) and p ∈ P (λ−1 ). Thus y = up · y1 = u · y2
min
min
where y2 = p · y1 ∈ Yβmin . Thus y2 = u−1 · y ∈ Yβmin ∩ Y−β
= Zβmin = Z−β
(notice that u ∈ U (λ) and y ∈ Yβmin implies u−1 · y ∈ Yβmin ). Therefore
y = u · y2 ∈ Uβ Zβmin . This implies that
Yβmin − U (λ)Zβmin ⊂ X s (l− ).
The other claim can be proved similarly.
¤
Proposition 8.2.4. 4.2.5 Proposition. Keep the notation of 4.2.2. For any β ∈
{β1 , · · · , βp }, let
p+ : Yβmin → Zβmin
min
min
p− : Y−β
→ Z−β
be the two natural projections. The subgroup of G that preserves the fiber of p±
over z ∈ Zβmin is Gz · U (λ).
Proof. Let’s prove it only for p+ . By 2.4.6 (ii) we need only to consider
min
elements of P (λ). Let p ∈ P (λ) and x ∈ p−1
) such that p · x ∈
+ (z) (z ∈ Zβ
−1
p+ (z). Then we have limt→0 λ(t) · x = z and limt→0 λ(t)p · x = z. But
limt→0 λ(t) · px = limt→0 λ(t) · p · λ−1 (t)λ(t)x = p0 · z where p0 = limt→0 λ(t) ·
p · λ−1 (t).
Hence p0 z = z which shows that p0 ∈ Gz . Hence limt→0 λ(t) · (p0 )−1 p ·
λ−1 (t) = id. That is, (p0 )−1 p ∈ U (λ), namely, p ∈ p0 U (λ) ⊂ Gz U (λ), which
implies the claim.
¤
±
Following the above proposition, denote p−1
± (z) by V . In what follows,
+
we will concentrate on V . The other can be treated similarly.
Now, the group Gz acts on U (λ) by conjugation. Thus for any u · z ∈
U (λ) · z and g ∈ Gz we have g · (u · z) = gug −1 g · z = gug −1 · z. This
shows that the orbit U (λ) · z is Gz -invariant. Let us identify V + with the
affine space Cn taking the point z as the origin. The group Gz acts on V +
117
linearly and defines a positive grading on the ring of regular functions
O(V + ) ∼
= C[T1 , . . . , Tn ]. We assume that each coordinate function Ti is
homogeneous of some degree qi > 0. Let R be the closure of the orbit
U (λ) · z. Since it is Gz -invariant it can be given by a system of equations
F1 = . . . = Fk = 0 where Fi are (weighted) homogeneous generators of
the ideal of regular functions on V + vanishing on R. We can write Fi =
Li (T1 , . . . , Tn )+Gi (T1 , . . . , Tn , where Li is a linear function in T1 , . . . , Tn and
Gi is a sum of (ordinary) homogeneous polynomials of degree > 1. Since R
is nonsingular at the origin, we can replace Fi by their linear combinations
to assume that there exists 1 ≤ s1 < . . . < sr ≤ n, r = codim R such that
X
Li = Tsi +
aij Tj , i = 1, . . . , r, Li = 0, i > r.
j6=s1 ,...,sk
Let W be the linear subspace of V + defined by the equations Tj = 0, j 6=
s1 , . . . , sr . It is obviously Gz -invariant. Consider the action map: a : U (λ)×
W → V +.
Lemma 8.2.5. 4.2.6 Lemma. The map a is an isomorphism.
Proof. First we claim that W ∩ R = {0}. In fact since the equations Fi
are weighted homogeneous each Gi with i ≤ r is a polynomial in Tj , j 6∈
{s1 , . . . , sr }. Thus, plugging all such Tj in Fsi we get Tsi = 0. This shows
that R ∩ W is given by the equations Ti = 0, i = 1, . . . , n. This proves
the claim. This also implies that the tangent space at the origin of V + is
equal to the direct sum of the tangent spaces of R and W . Now the source
V := U (λ) × W and the target V + are affine spaces of the same dimension.
The restriction of the map to U (λ) × {0} and to 1 × W are isomorphisms
onto their images R and W , respectively. Thus the differential of the map
a at the point (1, 0) is bijective. Now
a−1 (a(1, 0)) = a−1 (z) = {(u, w) ∈ U (λ) × W : w = u−1 · z}
consists of only one point (1, z). We know that the map is Gz -equivariant
and the action of Gz ∼
= C∗ on V and on V + is a good C∗ -action, i.e there is a
unique point in the closure of each orbit (the point (1, 0) for the first space
and the point z for the second one). As is well-known, a good C∗ -action
on an affine variety X is equivalent to a grading O(X) = ⊕i∈Z O(X)i of its
ring of regular functions satisfying O(X)i = {0} for i < 0 and O(X)0 = C.
The maximal ideal mX = ⊕i>0 O(X)i corresponds to the point lying in the
closure of all orbits. Returning to our situation we obtain that the map
a is defined by a homomorphism of graded rings a∗ : O(V + ) → O(V ).
The property that a is étale over (1, z) and a−1 (z) = (1, z) imply that
a∗ (mV + )O(V ) = mV . Since for any X as above, O(X) is generated as
C-algebra by mX ([Bo], Chapter III,§1, Proposition 1) we obtain that the
homomorphism a∗ is surjective. Since both rings are integral domains of
118
the same dimension this implies that the map a∗ is an isomorphism. Hence
a is an isomorphism.
¤
Theorem 8.2.6. Let G be a reductive algebraic group acting on a nonsingular
projective variety X. Let (C + , C − ) be a pair of chambers relevant to a truly
faithful cell F . Then, there are two birational morphisms
f + : X s (C + )//G → X ss (F )//G
and
f − : X s (C − )//G → X ss (F )//G
so that by setting Σ0 to be (X ss (F ) \ X s (F ))//G, we have
(i) f + and f − are isomorphisms over the complement to Σ0 ;
(ii) over each connected component Σ00 of Σ0 , the fibres of the maps f ± are weighted
projective spaces of dimension d± with the same weights;
(iii) d+ + d− + 1 = codim Σ00 .
Proof. Statement (i) follows immediately from Lemma 4.2.1.
To show (ii) and (iii), we apply the assertions and the notations of Lemmas 4.2.3 and 4.2.4.
(ii) It suffices to consider any particular stratum Sβ . Let λ be the corresponding one-parameter subgroup and
p+ : Yβmin → Zβmin
min
min
p− : Y−β
→ Z−β
be the two natural projections. Note that for any point z ∈ Zβmin , Gz =
λ(C∗ ).
From Lemma 4.2.4, one sees that the fiber of f + (resp. f − ) is just the
−1
−1
fiber p−1
− (z) (resp. p+ (z)) with U (λ) · z (resp. U (λ ) · z) removed modulo
−1
Gz ·U (λ) (resp. Gz ·U (λ )). We want to conclude that over each connected
component of
Σ0 = GZβmin //G = X sss (L0 )//G
f + and f − are weighted projective bundles.
±
±
±
Let us denote the fiber p−1
(not
± (z) by V . The group U (λ ) acts on V
+
−
linearly!), Gz acts on V (resp. on V ) linearly with positive (resp. negative) weights.
We need only consider V + ; the other set can be treated similarly.
Note that from Lemma 4.2.6 the orbit space of U (λ) on V + − U (λ) · z can
be identified with W − {z}, This implies that the quotient of V + − U (λ) · z
by Gz U (λ) can be identified with the quotient of W − {0} by Gz which is
a weighted projective space. Thus we have proved that the fiber of f − is a
weighted projective space.
119
Identical arguments can be applied to obtain a similar result for the map
f + . This finishes the proof of (ii).
(iii) We first claim that for any z ∈ Zβmin {g ∈ G|gz ∈ Zβmin } = Nλ , where
Nλ is the normalizer of Gz . To see this, if g · z ∈ Zβmin for some z ∈ Zβmin
then the isotropy λ of z should equal to the isotropy gλg −1 of g · z because
all of the elements of Zλmin have the same stabilizer Gz . This implies that
g ∈ Nλ . In fact, Zβmin is just the set of semi-stable points of the action of Nλ0
on Zβ where Nλ0 = Nλ /Gz .
By the above claim and Theorem 2.4.6 (ii), we have
Nλ = {g ∈ G|gz ∈ Zβmin } ⊂ {g ∈ G|gz ∈ Yβmin } = P (λ).
That is, Nλ is a subgroup in P (λ). So any element n of Nλ can be written
as ul with u ∈ U (λ) and l ∈ L(λ). Now L(λ) ⊂ Nλ because L(λ) is the
centralizer of λ, so u ∈ Nλ . We claim
L that the Lie algebra of U (λ) ∩ Nλ
must be zero. By 1.2.1, Lie(U (λ)) = hλ,αi>0 Lie(G)α . Choose a generator
h of the Lie algebra λ such
L that < λ, α >=< h, α > for all α. Take any
non-zero element X =
hλ,αi>0 Xα in Lie(U (λ)).LIf X were in Lie(Nλ ),
/
then we would have [h, X] ∈ Lie(λ). But [h, X] = hλ,αi>0 < h, α > Xα ∈
Lie(λ). Thus, U (λ) ∩ Nλ is finite. Or, L(λ) has finite index in Nλ . In fact,
what we will really need from this is just the following simple corollary:
dimNλ = dimL(λ).
To complete (iii), let Σ00 = GZβmin //G = Zβmin //Nλ0 . Then by the BialynickiBirula decomposition theorem ([B-B2]), we have
min
dim Zβmin + dim X = dim Yβmin + dim Y−β
.
Therefore,
dim X − dim Zβmin = dim (fibre of p+ ) + dim (fibre of p− ).
Hence
dim X − dim G − (dim Zβmin − dim Nλ0 )
= dim (fibre of p+ ) − dim U (λ) + dim (fibre of p− ) − dim U (λ−1 ) − 1.
because dim G = dim L(λ) + dim U (λ) + dim U (λ−1 ). Now it follows that
codim Σ00 = d+ + d− + 1.
¤
4.2.8 Remark. In the proof of (iii), one really should treat connected components of Zβmin separately since they may have different dimensions. However this does not affect the proof at all (cf. 4.1.2).
4.2.9 Remark. Theorem 4.2.7 can be generalized to the case of a faithful
(but not necessary truly faithful) wall. In this case, we can still define
a map a : U (λ) × W → V + which is equivariant with respect to the onedimensional radical of Gz . We obtain that W \{z} is contained in the union
of X s (l− ) and some strata Sαi . Thus the orbit space V + ∩ X s (l− )/Gz U (λ)
120
can be identified with the orbit space (W \ {z}) ∩ X s (l− )/Gz . This is what
the fibres of f − look like in this case. In the example communicated to us
by C. Walter, Gz is isomorphic to GL(n) and the fibres are isomorphic to
Grassmannians.
Exercises
121
9. G EOMETRIC I NVARIANT T HEORY AND B IRATIONAL G EOMETRY
Much of the materials here are taken from [39].
In this chapter, we prove that any two projective varieties with finite
quotient singularities related by an arbitrary birational morphism can be
realized as two geometric GIT quotients of a non-singular projective variety by a reductive algebraic group. Then, by applying the theory of Variation of Geometric Invariant Theory Quotients ([17]), we show that they
are related by a sequence of GIT wall-crossing flips.
9.1. GIT realization of Birational morphism.
Theorem 9.1.1. Let φ : X → Y be a birational morphism between two projective varieties with at worst finite quotient singularities. Then there is a smooth
polarized projective (GLn ×C∗ )-variety (M, L) such that
(1) L is a very ample line bundle and admits two (general) linearizations L1
and L2 with M ss (L1 ) = M s (L1 ) and M ss (L2 ) = M s (L2 ).
(2) The geometric quotient M s (L1 )/(GLn ×C∗ ) is isomorphic to X and the
geometric quotient M s (L2 )/(GLn ×C∗ ) is isomorphic to Y .
(3) The two linearizations L1 and L2 differ only by characters of the C∗ -factor,
and L1 and L2 underly the same linearization of the GLn -factor. Let L be
this underlying GLn - linearization. Then we have M ss (L) = M s (L).
By the construction of [?] (cf. §2 of [?]) , there is a polarized C∗ - projective
normal variety (Z, L) such that L admits two (general) linearizations L1
and L2 such that
(1) Z ss (L1 ) = Z s (L1 ) and Z ss (L2 ) = Z s (L2 ).
(2) C∗ acts freely on Z s (L1 ) ∪ Z s (L2 ).
(3) The geometric quotient Z s (L1 )/C∗ is isomorphic to X and the geometric quotient Z s (L2 )/C∗ is isomorphic to Y .
The construction of Z is short, so we reproduce it here briefly. Choose
an ample cartier divisor D on Y . Then there is an effective divisor E on
X whose support is exceptional such that φ∗ D = A + E with A ample on
X. Let C be the image of the injection N2 → N 1 (X) given by (a, b) →
aA + bE. The edge generated by φ∗ D divides C into two chambers: the
subcone C1 generated by A and φ∗ D, and the subcone C2 generated by
φ∗ D and E. The ring R = ⊕(a,b)∈N2 H 0 (X, aA + bE) is finitely generated
and is acted upon by (C∗ )2 with weights (a, b) on H 0 (X, aA + bE). Let
Z = Proj(R) with R graded by total degree (a + b). Then a subtorus C∗
of (C∗ )2 complementary to the diagonal subgroup ∆ acts naturally on Z.
The very ample line bundle L = OZ (1) has two linearizations L1 and L2
descended from two interior integral points in the chambers C1 and C2 ,
122
respectively. One verifies (1), (2) by algebra, and (3) by algebra and the
projection formula.
9.2. Partial desingularization of singular quotient.
9.3. Weak Factorization Theorem.
As a consequence, we obtain
Theorem 9.3.1. Let X and Y be two birational projective varieties with at worst
finite quotient singularities. Then Y can be obtained from X by a sequence of GIT
weighted blowups and weighted blowdowns.
The factorization theorem for smooth projective varieties was proved
by Wlodarczyk and Abramovich-Karu-Matsuki-Wlodarczyka a few years
ago ([3], [91], [92]). Hu and Keel, in [?], gave a short proof by interpreting it as VGIT wall-crossing flips of C∗ -action. My attention to varieties
with finite quotient singularities was brought out by Yongbin Ruan. The
proof here uses the same idea of [?] coupled with a key suggestion of Dan
Abramovich which changed the route of my original approach. Only the
first paragraph of §2 uses a construction of [?] which we reproduce for
completeness. The rest is independent. Theorem 9.1.1 reinforces the philosophy that began in [?]: Birational geometry of Q-factorial projective varieties is a special case of VGIT.
Proof. We continue from the proof of Theorem ??
Now, since C∗ acts freely on Z s (L1 ) ∪ Z s (L2 ), we deduce that Z s (L1 ) ∪
Z s (L2 ) has at worse finite quotient singularities. By Corollary 2.20 and
Remark 2.11 of [20], there is a smooth GLn - algebraic space U such that the
geometric quotient π : U → U/ GLn exists and is isomorphic to Z s (L1 ) ∪
Z s (L2 ) for some n > 0. Since Z s (L1 ) ∪ Z s (L2 ) is quasi-projective, we see
that so is U . In fact, since Z s (L1 ) ∪ Z s (L2 ) admits a C∗ -action, all of the
above statements can be made C∗ -equivariant. In other words, U admits
a GLn ×C∗ action and a very ample line bundle LU = π ∗ (Lk |Z s (L1 )∪Z s (L2 ) )
(for some fixed sufficiently large k) with two (GLn ×C∗ )- linearizations LU,1
and LU,2 such that
(1) U ss (LU,1 ) = U s (LU,1 ) and U ss (LU,2 ) = U s (LU,2 ).
(2) The geometric quotient U s (LU,1 )/(GLn ×C∗ ) is isomorphic to X and
the geometric quotient U s (LU,2 )/(GLn ×C∗ ) is isomorphic to Y . Moreover,
(3) the two linearizations LU,1 and LU,2 differ only by characters of the
C∗ factor.
Since we assume that LU is very ample, we have an (GLn ×C∗ )- equivariant embedding of U in a projective space such that the pullback of
123
O(1) is LU . Let U be the compactification of U which is the closure of U in
the projective space. Let LU be the pullback of O(1) to U . This extends LU
and in fact extends the two linearizations LU,1 and LU,2 to LU ,1 and LU ,2 ,
respectively, such that
ss
s
ss
s
U (LU ,1 ) = U (LU ,1 ) = U ss (LU,1 ) = U s (LU,1 )
and
U (LU ,2 ) = U (LU ,2 ) = U ss (LU,2 ) = U s (LU,2 ).
s
It follows that the geometric quotient U (LU ,1 )/(GLn ×C∗ ) is isomorphic
s
to X and the geometric quotient U (LU ,2 )/(GLn ×C∗ ) is isomorphic to Y .
Resolving the singularities of U , (GLn ×C∗ )-equivariantly, we will obs
s
tain a smooth projective variety M . Notice that U (LU ,1 ) ∪ U (LU ,2 ) =
U s (LU,1 ) ∪ U s (LU,2 ) ⊂ U is smooth, hence we can arrange the resolution so
that it does not affect this open subset. Let f : M → U be the resolution
morphism and Q be any relative ample line bundle over M . Then, by the
relative GIT (Theorem 3.11 of [34]), there is a positive integer m0 such that
for any fixed integer m ≥ m0 , we obtain a very ample line bundle over M ,
L = f ∗ Lm
⊗ Q, with two linearizations L1 and L2 such that
U
s
(1) M ss (L1 ) = M s (L1 ) = f −1 (U (LU ,1 )) and M ss (L2 ) = M s (L2 ) =
s
f −1 (U (LU ,2 )).
s
(2) The geometric quotient M s (L1 )/(GLn ×C∗ ) is isomorphic to U (LU ,1 )/(GLn ×C∗ )
which is isomorphic to X, and, the geometric quotient M s (L2 )/(GLn ×C∗ )
s
is isomorphic to U (LU ,2 )/(GLn ×C∗ ) which is isomorphic to Y .
Finally, we note from the construction that the two linearizations L1 and
L2 differ only by characters of the C∗ -factor, and L1 and L2 underly the
same linearization of the GLn -factor. Let L be this underlying GLn - linearization. It may happen that M ss (L) 6= M s (L). But if this is the case,
we can then apply the method of Kirwan’s canonical desingularization
([?]), but we need to blow up (GLn ×C∗ )-equivarianly instead of just GLn equivariantly. More precisely, if M ss (L) 6= M s (L), then there exists a reductive subgroup R of GLn of dimension at least 1 such that
MRss (L) := {m ∈ M ss (L) : m is fixed by R}
is not empty. Now, because the action of C∗ and the action of GLn commute, using the Hilbert-Mumford numerical criterion (or by manipulating
invariant sections, or by other direct arguments), we can check that
C∗ M ss (L) = M ss (L),
in particular,
C∗ MRss (L) = MRss (L).
124
Hence, we have
(GLn ×C∗ )MRss = GLn MRss ⊂ M \ M s (L).
Therefore, we can resolve the singularities of the closure of the union
of GLn MRss in M for all R with the maximal r = dim R and blow M
up along the proper transform of this closure. Repeating this process
at most r times gives us a desired nonsingular (GLn ×C∗ )-variety with
GLn -semistable locus coincides with the GLn -stable locus (see pages 157158 of [73]). Obviously, Kirwan’s process will not affect the open subset
M ss (L1 ) ∪ M ss (L2 ) = M s (L1 ) ∪ M s (L2 ) ⊂ M s (L). Hence, this will allow us
to assume that M ss (L) = M s (L).
This completes the proof of Theorem 9.1.1.
¤
The proof implies the following
Corollary 9.3.2. Let φ : X → Y be a birational morphism between two projective
varieties with at worst finite quotient singularities. Then there is a polarized
projective C∗ -variety (M , L) with at worst finite quotient singularities such that
X and Y are isomorphic to two geometric GIT quotients of (M , L) by C∗ .
9.4. Proof of Theorem 9.3.1.
Proof. Let φ : X −−− > Y be the birational map. By passing to the (partial)
desingularization of the graph of φ, we may assume that φ is a birational
morphism. This reduces to the case of Theorem 9.1.1.
We will then try to apply the proof of Theorem 4.2.7 of [17] (see also
[83]). Unlike the torus case for which Theorem 4.2.7 applies almost automatically, here, because (GLn ×C∗ ) involves a non-Abelian group, the
validity of Theorem 4.2.7 must be verified.
From the last section, the two linearizations L1 and L2 differ only by
characters of the C∗ -factor, and L1 and L2 underly the same linearization
of the GLn -factor. We denote this common GLn -linearized line bundle
by L. For any character χ of the C∗ factor, let Lχ be the corresponding
(GLn ×C∗ )-linearization. Note that Lχ also underlies the GLn -linearization
L. From the constructions of the compactification U and the resolution M ,
we know that M ss (L) = M s (L). In particular, GLn acts with only finite
isotropy subgroups on M ss (L) = M s (L). Now to go from L1 to L2 , we
will (only) vary the characters of the C∗ -factor, and we will encounter a
“wall” when a character χ gives M ss (Lχ ) \ M s (Lχ ) 6= ∅. In such a case,
since M ss (Lχ ) ⊂ M ss (L) = M s (L) which implies that GLn operates on
M ss (Lχ ) with only finite isotropy subgroups, the only isotropy subgroups
of (GLn ×C∗ ) of positive dimensions have to come from the factor C∗ , and
hence we conclude that such isotropy subgroups of (GLn ×C∗ ) on M ss (Lχ )
125
have to be one-dimensional (possibly disconnected) diagonalizable subgroups. This verifies the condition7 of Theorem 4.2.7 of [17] and hence its
proof goes through without changes.
¤
9.5. GIT on Projective Varieties with Finite Quotient Singularites. The
proof in §2 can be modified slightly to imply the following.
Theorem 9.5.1. Assume that a reductive algebraic group G acts on a polarized projective variety (X, L) with at worst finite quotient singularities. Then
there exists a smooth polarized projective variety (M, L) which is acted upon by
(G × GLn ) for some n > 0 such that for any linearization Lχ on X, there is a corresponding linearization Lχ on M such that M ss (Lχ )//(G × GLn ) is isomorphic
to X ss (Lχ )//G. Moreover, if X ss (Lχ ) = X s (Lχ ), then M ss (Lχ ) = M s (Lχ ).
This is to say that all GIT quotients of the singular (X, L) (L is fixed) by
G can be realized as GIT quotients of the smooth (M, L) by G × GLn . In
general, this realization is a strict inclusion as (M, L) may have more GIT
quotients than those coming from (X, L).
When the underlying line bundle L is changed, the compatification U
is also changed, so will M . Nevertheless, it is possible to have a similar construction to include a finitely many different underlying ample line
bundles. However, Theorem 9.5.1 should suffice in most practical problems because: (1) in most natural quotient and moduli problems, one only
needs to vary linearizations of a fixed ample line bundle; (2) Variation of
the underlying line bundle often behaves so badly that the condition of
Theorem 4.2.7 of [17] can not be verified.
7
Theorem 4.2.7 of [17] assumes that the isotropy subgroup corresponding to a wall is
a one-dimensional (possibly disconnected) diagonalizable group. The main theorems of
[83] assume that the isotropy subgroup is C∗ (see his Hypothesis (4.4), page 708).
126
10. C HOW Q UOTIENT
Much of the materials here are taken from [38]
10.1. Chow Variety.
We recall briefly the Chow variety. The reference is [21]. For a full account, see [63].
A k-dimensional
algebraic cycle in Pn is a formal finite linear combiP
nation C = i mi Ci with non-negative integer coefficients, where Ci are
k-dimensional irreducible closed subvarieties in Pn . The degree of C is
P
n
i mi deg(Ci ). Assume Y ⊂ P is a k-dimensional irreducible subvariety
of degree d. Consider the set Z(Y ) of all (n − k − 1)-dimensional linear
subspaces of Pn that intersect Y . Then Z(Y ) is an irreducible hypersurface
of degree d in the Grassmannian Gr(n − k − 1, Pn ). Let
B = ⊕m Bm = ⊕m H 0 (Gr(n − k − 1, Pn ), O(m))
be the coordinate ring of Gr(n − k − 1, Pn ). Then Z(X) is defined by the
vanishing of some element RY ∈ Bd which is unique up to homothety
(multiplication by a nonzero
P scalar). This element will be called the Chow
form of Y . Now let C = i mi Ci be an algebraic cycle of degree d. We
define the Chow form of C as
Y
RC =
RCmii ∈ Bd .
i
Let Chowd (Pn ) be the set of all k-dimensional algebraic cycles of degree d
in Pn . Then the map
Chowd (Pn ) → Proj(Bd ), C → RC
defines an embedding of Chowd (Pn ) into the projective space Proj(Bd ).
The variety Chowd (Pn ) with the projective algebraic structure defined by
the above embedding is called the Chow variety of k-dimensional algebraic cycles of degree d in Pn .
For a general projective variety X, an algebraic cycle is defined in the
same way as a non-negative linear combination of irreducible subvarieties. Each irreducible subvariety represents a homology class. The homology class of an algebraic cycle is defined as the induced sum of homology classes of the irreducible components. Now fix a homology class
δ. Let Chowδ (X) be the set of all algebraic cycles of X representing the
homology class δ. We can embed X into some projective space Pn , and
then cycles of X of homology class δ become cycles of Pn of degree d for
some d. Hence Chowδ (X) is embedded into Chowd (Pn ), and the variety
Chowδ (X) with the induced projective algebraic structure is called Chow
variety of algebraic cycles of X of homology class δ. (See [63] for further
details.)
127
Exercises
1. Let C∗ act on Pn by
t · [z0 , · · · , zn ] = [tw1 z0 , · · · , twn zn ].
Without loss of generality, we may arrange that
w0 ≤ · · · ≤ wn .
Prove that the action is generically free if and only if the integers in
{wi − w0 |wi − w0 6= 0}
are coprime.
2. Let C∗ act on Pn as in Exercise 1. Let C be the curve C∗ · [z0 , · · · , zn ]. The
degree of the curve C is given by
max{wi − w0 |zi 6= 0, wi − w0 6= 0}
.
g.c.d{wi − w0 |zi 6= 0, wi − w0 6= 0}
In particular, the generic orbit closures have the same degree.
3. Let G be a reductive group and V is a G-module. Show that the generic
orbit closures of the group G in P(V ) have the same degree.
128
10.2. Definition of Chow quotient.
Consider a reductive algebraic group action on a projective variety over
the field of complex numbers
G × X → X,
besides Mumford’s GIT quotient, Kapranov-Sturmfels-Zelevensky ([52])
for toric varieties and Kapranov ([51]) in general, considered the canonical
Chow quotient. Chow quotient and Hilbert quotient were also studied in
[33] at about the same time.
Definition 10.2.1. Let x0 be a fixed generic point of X and δ be the induced
homology class represented by the cycle G · x0 . There is a Zariski open
subset, U ⊂ X, containing the point x0 , such that G · x represents the same
homology class δ for all x ∈ U . Let Chowδ (X) be the component of the
Chow variety of X containing cycles of homology class δ. Then there is an
embedding
ι : U/G → Chowδ (X)
[G · x] → [G · x] ∈ Chowδ (X).
The Chow quotient, denoted by X//ch G, is defined to be the closure of
ι(U/G).
This definition is independent of the choice of the Zariski open subset
U . Note that the group G acts on the Chow variety Chowδ (X) by moving the Chow cycles, and under this action, the Chow quotient ι(U/G) is
contained in the fixed point set.
Remark 10.2.2. By our assumption of the action, there is a Zariski open
subset U0 such that points of U0 are isotropy-free. Let U be the Zariski open
subset in Definition 10.2.1. Now, we point out that, in this paper, unless
otherwise indicated, by generic points we shall always mean points in the
Zariski open subsets U0 ∩ U .
129
10.3. The Chow family and Chow cycles.
Let
F ⊂ X × (X//ch G)
be the family of algebraic cycles over the Chow quotient X//ch G defined
by
F = {(x, Z) ∈ X × (X//ch G)|x ∈ Z}.
Then, we have a diagram
F


fy
ev
−−−→ X
X//ch G
where ev and f are the projections to the first and second factor, respectively. For any point q ∈ X//ch G, we will call the fiber, f −1 (q), the Chow
fiber over the point q, and sometimes denote it by F (q).
Let G be a torus. Then by [1], for a generic point x ∈ X, we have
ΦL (G · x) = ΦL (X).
That is, for a generic point q ∈ X//ch G, we have
ΦL (ev(F (q))) = ΦL (X).
In fact, this is true for any point p ∈ X//ch G.
Proposition 10.3.1. Let G be a complex torus. Then for any point p ∈ X//ch G,
we have ΦL (ev(F (p))) = ΦL (X).
Proof. This is proved in Theorem (0.5.1) of [5]. We give slightly different
arguments. By replacing L by a large tensor power, we may assume that all
polytopes of the form ΦL (G · x) (for some x ∈ X) are lattice polytopes (see
Mumford’s Appendix to [75]). Let p ∈ X//ch G be an arbitrary point and
q ∈ X//ch G be a nearby generic point. Since p and q are nearby points, we
have that ev(F (p)) and ev(F (q)) are nearby in the Hausdorff topology on
the compact subsets of X. Hence by the continuity of ΦL , the two compact
lattice polytopes ΦL (ev(F (p))) and ΦL (ev(F (q))) are also nearby. Therefore,
they must be equal.
¤
P
Each Chow fiber F (q) as an algebraic cycle is of the form i mi G · xi
where mi are nonnegative integers and G · xi are orbits of the top dimension. We will call ∪i G · xi the support of F (q) and denote it by |F (q)|. Note
that ΦL (ev(F (q))) = ΦL (|F (q)|).
Observe from Proposition 10.3.1 that ΦL (|F (q)|) gives a subdivision of
the polytope P by the subpolytopes ΦL (G · xi ), any two of which either do
not meet or intersect along a proper face (see Theorem (0.5.1) of [5]).
130
We will make the following technical assumption which says that there
are no two Chow fibers with exactly the same support but different multiplicities. This assumption is needed for most of the paper except §3.6
(Perturbing, translating, and specializing) which is not affected.
Assumption 10.3.2. If p 6= q, then |F (p)| 6= |F (q)|.
We do not know whether this assumption always holds, nor do we
know an example violating the assumption. When X = Pn and G is a
subtorus
of the dense open torus (C∗ )n ⊂ Pn , if a Chow fiber F (q) =
P
i mi G · xi is a minimal toric degeneration of the generic orbit closures
(also called extreme toric degeneration), then mi is uniquely computed by
the volume of certain simplex of dim G (Theorem 3.2, Chapter 8, [21]. See
also [52]). Hence, it seems that the assumption holds in this case.
131
10.4. Chow quotient dominates GIT quotients.
Lemma 10.4.1.
Before we proceed further, two conventions are needed.
P
(1) The symbols [G · x] or [ i G · xi ] will be used to denote a point in
the Chow quotient X//ch G;
(2) while [G · x] will be used to denote a point in a given GIT quotient
X ss //G.
Theorem 10.4.2. For any linearize ample line bundle, we have a canonical birational morphism
πC : X//ch G → X ss (L)//G
Proof. We may assume that L is very ample and ΦL is the corresponding
moment map. Recall
X ss (L) = {x ∈ X|0 ∈ Φ(G · x)}.
For any q ∈ X//ch G, we have ΦL (ev(F (q))) = ΦL (X) and the moment map
images of the orbit closures in F (q) gives rise to a subdivision of ΦL (X)
by subpolytopes. Thus we have Hence F (q) ∩ X ss (L) contains a unique
polystable orbit, that is, the orbit whose moment map image contains 0.
This leads to a map
πC : X//ch G → X ss (L)//G
q −→ [F (q) ∩ X ss (L)]
which is, at least set-theoretically, well defined. First it is easy to see that
πC depends on q continuously. Second, πC , when restricted to the generic
orbit space U/G which is an open subset of both X//ch G and X ss (L)//G, is
of course isomorphism. Then by the removable singularity lemma 10.4.1,
πC is algebraic. Since both X//ch G and X ss (L)//G are projective, it is also
projective.
¤
It has been known that the Chow quotient dominates all GIT quotients
([51]). For any GIT quotient XC (§??), we shall use
πC : X//ch G → XC
to denote the corresponding canonical projection.
Remark 10.4.3. In fact, let U be any invariant open subset such that the
compact geometric quotient U/G exists. Then for any q ∈ X//ch G, F (q) ∩ U
is a single orbit in U (Theorem 0.4, [6]). In particular, let π : X//ch G → U/G
be the projection, then π(q) = [F (q) ∩ U ] ∈ U/G.
132
R EFERENCES
[1] M. F. Atiyah, Convexity and commuting Hamiltonians. Bull. London Math. Soc. 14
(1982), no. 1, 1–15.
[2] M. Atiyah and R. Bott, The moment map and equivariant cohomology. Topology 23
(1984), no. 1, 1–28. MR0721448.
[3] Dan Abramovich, Kalle Karu, Kenji Matsuki, Jaroslaw Wlodarczyk Torification
and Factorization of Birational Maps, J. Amer. Math. Soc. 15 (2002), no. 3, 531–572.
MR1896232, Zbl1032.14003.
[4] H. Bass and W. Haboush, Linearizing certain reductive group actions, Trans. Amer.
Math. Soc. 292 (1985), no. 2, 463–482.
[5] A. Bialynicki-Birula and A. Sommese, Quotients by C ∗ × C ∗ actions. Trans. Amer.
Math. Soc. 289 (1985), no. 2, 519–543.
[6] A. Bialynicki-Birula and A. Sommese, A conjecture about compact quotients by tori.
In: Complex Analytic Singularities. Adv. Studies in Pure Math. 8 (1986), 59–68.
[7] D. Birkes, Orbits of linear algebraic groups. Ann. of Math. (2) 93 (1971), 459–475.
MR0296077.
[8] A. Borel, Linear algebraic groups, Benjamin, New York, 1969.
[9] M Brion and C. Procesi, Action d’un tore dans une variété, in “Operator algebra, unitary
representations, enveloping algebras, and invariant theory”, Progress in Mathematics 192 (1990), Birkhäuser, 509-539.
[10] K. Behrend, Cohomology of Stacks, University of British Colombia (2002).
[11] A. Beilinson, J. Bernstein, and P. Deligne, Faisceaux Pervers. In: Analyse et topologie
sur les espaces singuliers, I. Astérisque #100, Soc. Math. de France (1983).
[12] H. Boden and Y. Hu, Variation of Moduli of Parabolic Bundles, Mathematische Annalen
Vol. 301 No. 3 (1995), 539 – 559.
[13] D. Burns, Y. Hu and T. Luo, HyperKähler manifolds and birational transformations in dimension 4, In: Vector bundles and representation theory, Contemporary Mathematics
Volume 322 (2003), 141 – 149.
[14] K. Cho, Y. Miyaoka, and N. Sheperd-Barron, Characterizations of projective space and
applications to complex symplectic manifolds. Higher dimensional birational geometry (Kyoto, 1997), 1–88, Adv. Stud. Pure Math., 35, Math. Soc. Japan, Tokyo, 2002.
MR1929792
[15] D. Cox, The homogeneous coordinate ring of a toric variety. J. Algebraic Geom. 4 (1995),
no. 1, 17–50.
[16] S. D. Cutkosky, Strong Toroidalization of Birational Morphisms of 3-Folds.
math.AG/0412497
[17] I. Dolgachev and Y. Hu, Variation of Geometric Invariant Theory Quotients, Publ. Math.
I.H.E.S. 87 (1998), 5 – 56. MR1659282, Zbl 1001.14018.
[18] S. Donaldson, Scalar curvature and projective embeddings. I, J. Differential Geom. 59
(2001), no. 3, 479–522.
[19] S. Donaldson, Scalar curvature and stability of toric varieties . J. Differential Geom. Volume 62 (2002), 289–349.
[20] D. Edidin, B. Hassett, A. Kresch, A. Vistoli, Brauer Groups and Quotient Stacks. Amer.
J. Math. 123 (2001), no. 4, 761–777. MR1844577, Zbl 1036.14001.
[21] I. Gelfand, M. Kapranov, and A. Zelevinsky, Discriminants, resultants, and multidimensional determinants. Mathematics: Theory & Applications (1994).
[22] V. Guillemin and S. Sternberg, Birational equivalence in the symplectic category. Invent.
Math. 97 (1989), no. 3, 485–522. MR1005004.
[23] M. Goresky and R. MacPherson, On the topology of torus actions, in “Algebraic
Groups” (1985).
161
[24] G. Faltings and G. Wüstholz, Diophantine approximations on projective spaces, Invent.
Math 116 (1994), 109–138.
[25] P. Foth and Y. Hu, Toric degenerations of weight varieties and applicationsmath, Travaux
Mathématiques (2005), math.AG/0406329.
[26] B. Fu, Birational geometry in codimension 2 of symplectic resolutions. math.AG/0409224
[27] W. Fulton and R. MacPherson, A compactification of configurations spaces, Ann. of
Math. 139 1994), 183-225.
[28] A. Givental, Equivariant Gromov-Witten invariants. Internat. Math. Res. Notices 1996,
no. 13, 613–663. MR1408320
[29] P. Hacking, Compact moduli of hyperplane arrangements, math.AG/0310479.
[30] R. Hartshorne, Algebraic Geometry, Springer-Verlag, 1977.
[31] S. Helgason, Differential geometry, Lie groups, and symmetric spaces. Pure and Applied Mathematics, 80. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR0514561
[32] Y. Hu, Geometry and topology of quotient varieties, Ph.d Thesis, MIT , 1991.
[33] Y. Hu, Geometry and topology of quotient varieties of torus actions, Duke Math. Journal
68 (1992), 151 – 183.
[34] Y. Hu, Relative geometric invariant theory and universal moduli spaces. Internat. J. Math.
7 (1996), no. 2, 151–181. MR1382720, Zbl 0889.14005.
[35] Y. Hu, Moduli Spaces of Stable Polygons and Symplectic Structures on M 0,n , Compositio
Mathematica Vol. 118 (1999), 159 – 187.
[36] Y. Hu, Combinatorics and Quotients of Toric Varieties, Discrete & Computational Geometry Vol. 28 (2002), 151 – 174.
[37] Y. Hu, Stable Configurations of Linear Subspaces and Vector Bundles. Quarterly Journal
of Pure and Applied Mathematics Volume 1 No. 1 (2005), 127 – 164.
[38] Y. Hu, Topological Aspects of Chow Quotients. Journal of Differential Geometry Volume
69 No. 3 (2005), 399-440.
[39] Y. Hu, Factorization Theorem for Projective Varieties with Finite Quotient Singularities,
Journal of Differential Geometry Volume 68 No. 3 (2004), 587 – 593.
[40] Y. Hu, M 0,n as Chow Quotient Revisited, In preparation.
[41] Y. Hu, Toric Degenerations of GIT Quotients, Chow Quotients, and M0,n . Asian Journal
of Mathematics (2006).
[42] Y. Hu, Quotients by reductive group, Borel subgroup, unipotent group and maximal torus.
Pure and Applied Mathematics Quarterly, Volume 2 No. 4 (2006), special issue in
honor of Robert MacPherson, 135 - 151.
[43] Y. Hu, Geometric Invariant Theory and Birational Geometry, Proceedings of International Congress of Chinese Mathematicians-2004, to appear (2005), 1 – 25.
[44] Y. Hu, In preparation.
[45] Y. Hu and S. Keel, Mori dream spaces and GIT, Dedicated to William Fulton on the
occasion of his 60th birthday. Michigan Math. J. 48 (2000), 331–348. MR1786494, Zbl
pre01700876.
[46] Y. Hu and S. Keel, A GIT proof of the weighted weak factorization theorem. unpublished
short notes, available at Math ArXiv, math.AG/9904146.
[47] D. Huybrechts, Compact HyperKähler manifolds: basic results, Invent. Math. 135 (1999),
63-113.
[48] Y. Hu and W. Li, Variation of the Gieseker and Uhlenbeck Compactifications, International
Journal of Mathematics Vol. 6 No. 3 (1995), 397 – 418.
[49] Y. Hu and S.-T. Yau, HyperKḧler Manifolds and Birational Transformations, Adv. Theor.
Math. Phys. Volume 6 Number 3 (2002), 557-576.
[50] Y. Hu, C.-H. Liu, and S.-T. Yau, Toric Morphisms and Fibrations of Toric Calabi-Yau
hypersufaces, Adv. Theor. Math. Phys. Volume 6 Number 3 (May 2002), 457 – 506.
162
[51] N. Iwahori and H. Matsumoto, On some Bruhat decomposition and the structure of Hecke
rings of p-adic Chevalley groups, Publ. Math. I.H.E.S. 25 (1965), 5 – 48.
[52] N. Jacobson, Basic Algebra II, Freeman and Company, 1980.
[53] M. Kapranov, Chow quotients of Grassmannian, I.M. Gelfand Seminar Collection, 29–
110, Adv. Soviet Math., 16, Part 2, Amer. Math. Soc., Providence, RI, 1993.
[54] M. Kapranov, B. Sturmfels and A. Zelevinsky, Quotients of toric varieties, Math. Ann.
290 (1991), no. 4, 643–655.
[55] S. Keel, Intersection theory of moduli spaces of stable pointed curves of genus zero, Transactions AMS. 330 (1992), 545–574.
[56] S. Keel and S. Mori, Quotients by groupoids. Ann. of Math. (2) 145 (1997), no. 1, 193–
213.
[57] S. Keel and J. Tevelev, Chow Quotients of Grassmannians II, math.AG/0401159.
[58] G. Kempf, Instability in invariant theory. Ann. of Math. (2) 108 (1978), no. 2, 299–316.
MR0506989
[59] G. Kempf and L. Ness, The length of vectors in representation spaces. Algebraic geometry (Proc. Summer Meeting, Univ. Copenhagen, Copenhagen, 1978), pp. 233–243,
Lecture Notes in Math., 732, Springer, Berlin, 1979. MR0555701.
[60] F. Kirwan, Cohomology of quotients in symplectic and algebraic geometry. Mathematical
Notes, 31. Princeton University Press, Princeton, NJ, 1984. MR0766741
[61] F. Kirwan, Partial desingularisations of quotients of nonsingular varieties and their Betti
numbers. Ann. of Math. (2) 122 (1985), no. 1, 41–85. MR0799252.
[62] A. Klyachko, Stable bundles, representation theory and Hermitian operators, Selecta
Math. 4 (1998), 419-445.
[63] J. Kollár, Projectivity of complete moduli, Journal of Diff. Geom. 32 (1990), 235 – 268.
[64] J. Kollár, Quotient spaces modulo algebraic groups. Ann. of Math. (2) 145 (1997), no. 1,
33–79.
[65] J. Kollár, Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer
Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics],
32. Springer-Verlag, Berlin, 1996.
[66] J. Kollár and S. Mori, Birational geometry of algebraic varieties, Cambridge University
Press, 1998.
[67] L. Lafforgue, Chirurgie des grassmanniennes. (French) [Surgery on Grassmannians]
CRM Monograph Series, 19. American Mathematical Society, Providence, RI, 2003.
[68] L. Lafforgue, Pavages des simplexes, schémas de graphes recollés et compactification des
/PGLr , Invent. Math. 136 (1999), no. 1, 233–271. Erratum: Invent. Math. 145
PGLn+1
r
(2001), no. 3, 619–620.
[69] Jun Li, Stable morphisms to singular schemes and relative stable morphisms, J. Diff. Geom.
57 (2001), no. 3, 509–578.
[70] Jun Li, A degeneration formula of GW-invariants, J. Diff. Geom. 60 (2002), no. 2, 199–
293.
[71] B, Liang, K. Liu and S. T. Yau, Mirror Principles, I Asian J. of Math. 1 (1997). no. 4,
729–763. MR1621573
[72] B, Lian, K. Liu and S. T. Yau, Mirror Principles, II, III, Asian J. of Math. (1999) no. 1,
109–146 and no. 4, 771–800, MR1701925 and MR1797578
[73] B, Lian, K. Liu and S. T. Yau, Mirror principle. IV Surveys in differential geometry,
433–474, Surv. Differ. Geom., VII, Int. Press, Somerville, MA, 2000.
[74] H.-Z. Luo, Geometric criterion for Gieseker-Mumford stability of polarized manifolds, Journal of Diff. Geom. 49 (1998), no. 3, 577–599.
[75] D. Mumford, Geometric Invariant Theory, 1962. Berlin, New York. MR1304906, Zbl
0797.14004.
163
[76] P. Newstead, Introduction to moduli problems and orbit spaces, Tata Institute, Bombay,
1978.
[77] L. Ness, Stratification of the null cone via the moment map. With an appendix by David
Mumford. Amer. J. Math. 106 (1984), no. 6, 1281–1329. MR0765581
[78] R. Pandharipande, A compactification over Mg of the universal moduli space of slopesemistable vector bundles, J. Amer. Math. Soc. 9 (1996), no. 2, 425–471. MR1308406
[79] Z. Reichstein, Stability and equivariant maps, Inven. Math. 96 (1989), 349-383.
[80] D.H. Phong and J. Sturm, Stability, Energy Functions, and Kähler-Einstein Metrics,
arXiv:math.DG/0203254.
[81] J. Ross and R. P. Thomas, A study of the Hilbert-Mumford criterion for the stability of
projective varieties. math.AG/0412519
[82] J. Ross and R. P. Thomas, An obstruction to the existence of constant scalar curvature
Kähler metrics, math.DG/0412518
[83] Y. Ruan, Stringy geometry and topology of orbifolds. Symposium in Honor of C. H.
Clemens (Salt Lake City, UT, 2000), 187–233, Contemp. Math., 312, Amer. Math. Soc.,
Providence, RI, 2002.
[84] C. Simpson, Moduli spaces of representations of the fundamental group of a smooth projective variety, I and II, Inst. Hautes Études Sci. Publ. Math. No. 79 (1994), 47–
129. MR1307297. Inst. Hautes Études Sci. Publ. Math. No. 80 (1994), 5–79 (1995).
MR1320603.
[85] M. Thaddeus, Geometric invariant theory and flips, J. Amer. Math. Soc. 9 (1996), no. 3,
691–723. MR1333296, Zbl 0874.14042.
[86] G. Tian, Kähler-Einstein metrics on algebraic manifolds, LNM 1646, Spinger, Berlin,
1996.
[87] G. Tian, Kähler-Einstein metrics with positive scalar curvature. Invent. Math. 130 (1997),
1–39.
[88] B. Totaro, Tensor products of semistables are semistable, Geometry and Analysis on complex manifolds, World Sci. Publ. 1994, pp. 242-250.
[89] X. W. Wang, Balanced point, stability and vector bundles over projective manifolds. Math.
Res. Lett. 9 (2002), no. 2-3, 393–411.
[90] A. Ulyanov, A polydiagonal compactification of configuration spaces, math.AG/9904049.
[91] E. Vieweg, Quasi-projective moduli of polarized manifolds, Springer-Verlag, 1995.
[92] J. Wierzba and J. Wiśniewski, Small contractions of symplectic 4-folds. Duke Math. J.
120 (2003), no. 1, 65–95. MR2010734.
[93] J. Wlodarczyk, Birational cobordisms and factorization of birational maps. J. Algebraic
Geom. 9 (2000), no. 3, 425–449. MR1752010, Zbl 1010.14002.
[94] J. Wlodarczyk, Toroidal varieties and the weak factorization theorem. Invent. Math. 154
(2003), no. 2, 223–331. MR2013783, Zbl pre02021105.
[95] J. Wlodarczyk, Decomposition of birational toric maps in blow-ups & blow-downs. Trans.
Amer. Math. Soc. 349 (1997), no. 1, 373–411. MR1370654
[96] S.-T. Yau, Open problems in geometry, Proc. Symp. Pure Math. 54 (1993) 1-28.
[97] S.-T. Yau, Mathematical aspects of string theory. Proceedings of the conference held at
the University of California, San Diego, California, July 21–August 1, 1986. Edited by
S.-T. Yau. Advanced Series in Mathematical Physics, 1. World Scientific Publishing
Co., Singapore, 1987. MR0915812
[98] S. Zhang, Heights and reductions of semi-stable varieties, Compositio Mathematica 104
(1996) 77-105.
164
D EPARTMENT
USA
OF
M ATHEMATICS , U NIVERSITY
OF
A RIZONA , T UCSON , AZ 86721,
E-mail address: yhu@math.arizona.edu
IMS, C HINESE U NIVERSITY OF H ONG K ONG , H ONG K ONG , C HINA
E-mail address: yhu@ims.cuhk.edu.hk
165