Hodge theory and boundaries
of moduli spaces
Phillip Griffiths∗
Based on correspondence and discussions with Mark Green, Colleen Robles and Radu Laza
∗
Institute for Advanced Study
1/82
I
I
I
I
The general theme of this talk is the use of Hodge theory
to study questions in algebraic geometry.
This is a vast and classical subject; here we shall focus on
one particular aspect.
In algebraic geometry a subject of central interest is the
study of moduli spaces M and their compactifications M.
Points in ∂M = M\M correspond to singular varieties X0
that arise from degenerations X → X0 of a smooth
variety X .
2/82
I
It is well known and classical that X0 may be “simpler”
than X and understanding X0 may shed light on X . For
an example, any smooth curve X may be degenerated to
a stick curve
X0 =
= configuration of P1 ’s
and deep questions about the geometry of a general X
may frequently be reduced to essentially combinatorial
ones on X0 .
3/82
I
The central thesis of this talk is
Hodge theory may be used to guide the study of
degenerations X → X0 .
Underlying this thesis are the points
I
I
I
the polarized Hodge structure on H ∗ (X , Z) (mod
torsion) is the basic invariant of a smooth projective
variety X ;
the study of degenerations (V , Q, F ) → (V , Q, W , Fo )
of polarized Hodge structures is fairly highly developed;
this understanding may be used to suggest what the
possible degenerations X → X0 should be and where to
look for them.
4/82
This thesis is well known and has been effectively used in the
classical case when the Hodge structures are parametrized by a
Hermitian symmetric domain and 1-parameter degenerations
are used. It has been utilized in the non-classical case mainly
for 1-parameter degenerations of Calabi-Yau threefolds, but to
my knowledge not so much in other non-classical cases or for
several parameter degenerations other than for the case of
algebraic curves.
5/82
I
the structure of this talk will be
I
I
I
discussion of some of the basic aspects of Hodge theory;
the example of algebraic curves, which indicates how the
study might go in the classical case;
discussion of the first non-classical case, namely that of
an algebraic surface X of general type with h2,0 (X ) = 2,
h1,0 (X ) = 0 and KX2 = 2; this is based on
correspondence with Mark Green, and also with Colleen
Robles and Radu Laza, and reflects developments in a
very early stage.
6/82
I
A final general point: a schematic of how Hodge theory
interacts with algebraic geometry for the above surface is
canonical
←→
H 0 (Ω2X ) = H 2,0 (X )
series |KX |
bi-canonical
variation
of
Hodge
structure
←→
series |2KX |
associated to H 1 (C ), C ∈ |KX |
..
.
which arises from
KX = KC
C
..
.
7/82
Definition
A polarized Hodge structure (PHS) of weight n is given by
(V , Q, F • ) where
I V is a Q-vector space (in practice we will have V ⊃ VZ );
I Q · V ⊗ V → Q is a non-degenerate form with
Q(u, v ) = (−1)n Q(v , u);
I F • is a decreasing filtration F n ⊂ F n−1 ⊂ · · · ⊂ F 0 = VC
satisfying
Fp ⊕ F
I
I
n−p+1 ∼
−
→ VC ,
0 5 p 5 n;
Q(F p , F n−p+1 ) = 0;
Q(v , C v̄ ) > 0 for v 6= 0, where C is the Weil operator.
8/82
q
Setting V p,q = F p ∩ F = V
q,p
, the third condition above is
VC = ⊕V p,q where V
p,q
= V q,p
and Cv = i p−q on V p,q .
Example
Let X ⊂ PN be a smooth projective variety of dimension d
and L ∈ H 2 (X , Q) the restriction of the generator of a
hyperplane class in H 2 (PN , Z). Then multiplication by L gives
a nilpotent operator on H ∗ (X , Q)
L : H m (X , Q) → H m+2 (X , Q).
9/82
The Hard Lefschetz theorem
∼
Lk : H m−k (X , Q) −
→ H m+k (X , Q)
implies that the action of L may be uniquely extended to the
action of an sl2 -triple {L, Y , Λ} where
Y = (m − d)Id on H m (X , Q)
=
is the grading. This sl2 -action decomposes H ∗ (X , Q) into
irreducibles whose basic building blocks are the lowest weight
spaces
n
o
Lk+1
H m−k (X , Q)prim = ker H m−k (X , Q) −−→ H m+k+2 (X , Q)
V
of the irreducible summands.
10/82
Setting Q(u, v ) = (Lk u · v )[X ] and using the decomposition
H n−k (X , C)prim induced by the Hodge decomposition
H m−k (X , C) =
⊕ H p,q (X ) gives a PHS of weight
p+q=m−k
m − k. Instead of the usual Hodge diamond, we will picture
the groups in the first quadrant in the (p, q) plane.
q
r
dim X = 1
L
r
r
p
↑
H
= holomorphic 1-forms
on the compact Rieman surface X .
0
(Ω1X )
11/82
dim X = 2 and X is regular (H 1,0 (X ) = 0)
r
r
0
LH (X ) H
H
j r r
H
H 1,1 (X )prim
r
H 0 (Ω2X )
<o>
12/82
dim X = 1 The PHS on H 1 (X ) for X a compact Riemann
surface is classically represented by the matrix of periods
ˆ
γ
ω,
···
δ,γ
, ω ∈ H 0 Ω1X
δ
Even more classically, if X is given as the Riemann surface
associated to an algebraic curve, e.g., the one of genus g
given in C2 by
2g +2
2
f (x, y ) = y −
Y
(x − ai ) = 0,
ai distinct
i=1
g (x) dx
2g (x) dx
ω=
=
,
y
fy (x, y )
0 5 deg g (x) 5 g − 1
13/82
the periods are
δ
a1
a2
a3
a4
...
γ
The closure in P2 of the above affine curve has a singular
point at [0, 0, 1] and the condition deg g (x) 5 g − 1 is the one
that gives a holomorphic 1-form on X .
14/82
dim X = 2 The surfaces we shall be considering are minimal,
of general type and have
h2,0 (X ) = 2,
h1,0 (X ) = 0,
KX2 = 2.
They are extremal; i.e., they have the maximum h2,0 (X )
subject to the other conditions above. They were analyzed
classically by Castelnuovo and more recently by Horikawa. A
general one is the minimal desingularization of the quartic
surface on P4 given in affine coordinates x1 , x2 , x3 , x4 by
(
f (x) = h(x) − k(x)2 = 0
g (x) = x12 − x2 = 0
where h(x), k(x) are a general cubic, quadric.
15/82
This surface has as singularity a double curve with eight pinch
points. Think of it as an analogue of the smooth curve with
affine equation
2g +2
Y
2
y −
(x − ai ) = 0
i=1
encountered above.
It is the first regular, non-classical extremal surface. We will
say more about it later.
16/82
The holomorphic 2-forms are
ω=
`(x)dx3 ∧ dx4
,
fx1 (x)gx2 (x)
deg `(x) 5 1
and where `(x) vanishes on the double conic so that ω is
holomorphic on X . With these conditions we find that
dx
where
h2,0 (X ) = 2. Think of ω as the analogue of gf(x)
y (x)
deg g (x) 5 g − 1 above. One has b2 (X ) = 32, so that
h1,1 (X )prim = 31 and the PHS is given by the matrix of
periods.
17/82
The surface is more conveniently given in the homogenous
coordinate [t0 , t1 , x3 , x4 ] in the weighted projective space
P(1, 1, 2, 2) by
t02 G (t02 , t0 t1 , t12 , x3 , x4 ) = F (t02 , t0 t1 , t12 , x3 , x4 )2 .
For fixed [t0 , t1 ] we obtain genus 3 plane quartic curves which
constitute the pencil |KX |.
<o>
18/82
Definition
D = period domain given by {(V , Q, F • ), dim V p,q = hp,q }
Ď = compact dual given by {(V , Q, F • ), Q(F p , F n−p+1 ) = 0}.
If G = Aut(V , Q), then
D = GR /H,
H compact
∩
Ď = GC /P,
P parabolic and H = P ∩ GR .
19/82
Example
D = Hg ∼
= {Z = g × g matrix, Z = t Z and Im Z > 0}
= Sp(2g , R)/U(g )
Ď = GrassL (g , C2g ) = Lagrangian g -planes in (C2g , Q)
0 I
where Q = −I
0 .
The embedding D ,→ Ď is
Z
Z → span of the columns of
.
Ig
For g = 1 we have H ,→ P1 and GR = SL2 (R) acting by linear
fractional transformations.
20/82
Example
Ď = GrassQ (2, VC ) = {2-planes E : Q(E , E ) = 0}
∪
D = 2-planes E with Q(E , E ) < 0.
The GR = SO(4, h) and D = GR /U(2) × SO(h) is not an
Hermitian symmetric domain (HSD).
Definition
A variation of Hodge structure (VHS), or period mapping, is
given by a holomorphically varying family of PHS’s
parametrized by a complex manifold S and satisfying a
differential constraint. In order to have the PHS’s occurring in
a fixed vector space (V , Q) we need to use the universal
e → S.
covering S
21/82
Example
e = H = {z : Im z > 0}, and
S = ∆∗ = {t : 0 < |t| < 1}, S
t = e 2πiz . We may take products of this to have S = ∆∗` .
Definition
A VHS is given by
e
S
S
/
e
Φ
Φ
/
D
Γ\D
where we have the monodromy representation ρ : π1 (S) → GZ
e is an equivariant holomorphic mapping satisfying
and where Φ
e
e
e
(i) Φ(γz)
= ρ(γ)Φ(z)
for γ ∈ π1 (S), z ∈ S;
(ii)
dFzp
dz
⊂ Fzp−1 (infinitesimal period relation = IPR).
22/82
Example 1
Q
S = ∆∗ and Xt = {y 2 − x(x − t) (x − ai ) = 0}. The family
X → ∆∗ is a differentiable fibre bundle, but we may only
identify the H 1 (Xt , Z) with Z2g up to the action of
monodromy when t turns around the origin
γ
γ
−→
···
···
δ
γ→γ+δ
0
···
· · · −→
t
δ
γ
γ
23/82
At t = 0 we have the singular curve
γ
···
δ
y2 = x2
Q
···
(x − ai )
The IPR is trivial in this case, as it is whenever the period
domain D is a Hermitian symmetric domain.
γ
δ
γ
···
···
δ
24/82
The pictures of the singularity in Example 1 are
←→
6
real picture
= neighborhood of the node
6
topological picture
25/82
Example 2
This is similar to Example 1, but it is harder to draw the
pictures. We will be interested in the degeneration of the
surface above, e.g., given by
f (x, t) = (x1 − t)q(x) − k(x)2 = 0
where q(x) is a general quadric. It is much harder to draw the
pictures.
26/82
The real picture of the singularity is
p
This is a double curve with a pinch point.
27/82
We remark that if we have a family of smooth projective
varieties
π
− ∆∗ ,
Xt = π −1 (t)
X∗ →
the limit
lim Xt
t→0
is not well defined — it depends on how we complete
X∗ → ∆∗ to X → ∆ to have X0 . Moreover, even if we have
conditions on X that uniquely define X0 , the geometry of X0
may be complicated. This situation is even more so when we
have a higher dimensional parameter space, e.g., S = ∆∗` .
This brings us to the main point of the talk.
28/82
I
I
Degenerating Hodge structures given by Φ : ∆∗` → Γ\D
are well understood when ` = 1 and a lot is known when
` > 1.
One may hope that this knowledge can be used to help
guide the geometric understanding of X → X0 .
We will try to (i) explain how this principle works for algebraic
curves (classical), (ii) explain the issues when X is a surface of
general type (like g = 2 for algebraic curves), and (iii) discuss
part of the one surface example where there are very early
indications of how the above philosophy might work out quite
nicely. Implementing this philosophy in the end of course
requires algebro-geometric arguments.
29/82
The analysis of degenerating PHS’s uses Lie theory and the
geometry of homogeneous complex manifolds. The starting
point is the
Monodromy Theorem
For Φ : ∆∗ → Γ\D the generator T ∈ Aut(VZ , Q) given by
the image of the identity in π1 (∆∗ ) is unipotent — i.e., the
eigenvalues of T are roots of unity.
Replacing t by t k we may assume that T = exp N where
N = log T = (T − I ) − 12 (T − I )2 + · · · ∈ Aut(VQ , Q) is
nilpotent. Moreover,
N n+1 = 0
where n = weight of the PHS’s.
30/82
Definition
Φ
A nilpotent orbit is a VHS ∆∗r −
→ {T k }\D given by
e
Φ(z)
= exp(zN) · F0
where F0 ∈ Ď and exp(zN) · F0 ∈ D for Im z > log(r ).
Rescaling t = λt we may assume r = 1 and F0 ∈ D. The IPR
is equivalent to
NF0p ⊂ F0p−1 .
Theorem (Schmid)
Any VHS over ∆∗ may be well approximated by a nilpotent
orbit.
31/82
Classically it was known (theorem on regular singular points)
that any period matrix may be expanded as a polynomial in
log t with holomorphic coefficients — the nilpotent orbit is
(with a suitable normalization) obtained by evaluating the
coefficients at t = 0. The “suitable normalization” is subtle
and is given a structure by the further approximating sl2 -orbit.
Example
For y 2 = x(x − t)(x − a)(x − b), complex analysis may be
used to show that
ˆ t
ˆ
dx
log t
dx
+ h(t),
=
= k(t),
k(0) =
6 0.
2πi
0 y
γ y
32/82
The nilpotent orbit is obtained by using h(0), k(0). Rescaling
t and dx/y , we may take h(0) = 0, k(0) = 1. The nilpotent
orbit is then
" #
z
(∗)
z→
∈ H ⊂ P1 .
1
Note that F0 =
0
1
∈ ∂H; this will be a general situation.
Definition
An sl2 -orbit is given by a representation ρ : SL2 (R) → GR with
ρ∗ ( 00 10 ) = N which induces an equivariant VHS
H ,→ D.
33/82
Theorem (Schmid)
A nilpotent orbit may be approximated by an sl2 -orbit.
In this way we may understand limt→0 Φ(t).
What is the limiting object?
To explain this, we note that any N may be graded by a
semi-simple Y with integral eigenvalues and weight spaces Vk
where −n 5 k 5 n, and we define the weight filtration
W (N)m = ⊕ Vk+n
k5m
W (N)0 ⊂ W (N)1 ⊂ · · · ⊂ W (N)2n = V .
With indices running opposite to the case of the hard
Lefschetz theorem we have
N : W (N)k → W (N)k−2 .
34/82
Definition
A mixed Hodge structure (MHS) is given by (V , W• , F ) where
W• is an increasing filtration on V and F is a decreasing
filtration on VC , and where F induces on each
GrW
k V = Wk /Wk−1 a Hodge structure of weight k.
Definition
A polarized limiting mixed Hodge structure (we shall just use
LMHS) (V , Q, W (N), F ) is given by a MHS (V , W (N), F )
such that for k = n the bilinear forms Qk (u, v ) = Q(N k u, v )
W (N)
induce PHS’s on Gk,prim V .
35/82
W (N)
Here the primitive spaces are defined on GrV
of a smooth projective variety with
(
N ↔L
Y ↔ Y,
as in the case
where as noted above N goes in the opposite direction to L.
The above definition depends only on N and not on the Y .
The sl2 -orbit theorem gives us a choice of Y .
36/82
The punch line of the above is: A VHS
Φ : ∆∗ → Γ\D
gives a LMHS
lim Φ(t) = (V , W (N), F0 ).
t→0
The choice of F0 is not unique because rescaling t → λt gives
F0 → exp(λN)F0 . It may be normalized so that (among other
things) F0 ∈ ∂D.
37/82
D
F∞
exp(iy
−1
+
N )F∞
F0
exp(iyN)F0
F∞ = lim exp(iyN)F0
y →∞
F0 = lim exp(iy −1 N + )F∞
y →0
where N + = ( 01 00 ). The point here is that a lot is known
about degeneration of PHS’s.
38/82
Example
When the weight n = 1 we have N 2 = 0 and the weight
filtration is
W0 (N) ⊂ W1 (N) ⊂ W2 (N) = V .
W0 (N) = Im N = isotropic subspace
(Q = 0 on it)
⊥
W1 (N) = Ker N = W0 (N) .
39/82
Elementary linear algebra shows that we may choose a
W (N)
symplectic basis such that V ∼
V and
= ⊕ Grk
k

0
0
I



Q =  0 Q0 0 ,
−I 0 0


0 0 A


N = 0 0 0  ,
Q0 =
0
Ig0
−Ig0
0
!
A = t A, A > 0.
0 0 0
40/82
Recalling that the period domain D = Hg = {g × g matrices
Z , Z = t Z and Im Z > 0}, the nilpotent orbit is
!
log t
A
W (N)
2πi
(LMHS).
,
Z0 ∈ Hg0 ∈ Gr1
Z (t) =
Z0
Taking A = I and having in mind
log t
period of the form
+ h(t)
2πi



acquiring a node as Xt → X0 plus

0
1
←→
,
having ωt ∈ H (ΩXt ) where




ω0 has a pole at the node
41/82
what is suggested is that we look for a curve X0 where the
e0 has genus g0 and where X0 has g1 = g − g0
normalization X
nodes; e.g.,
...
| {z }
g1
...
...
-
| {z }
g0
= X0
6
r
r
r
r
...
e0
=X
=
Xt =


curve of genus g0 with

g1 sets of pairs of

marked points pi , qi
42/82
Monodromy is
γi → γi + δi
where δi is the vanishing cycle at thei th node.
e0 we have differentials ωi ∈ H 0 Ω1 (pi + qi ) of the 3rd
On X
e0
X
kind with Respi ,qi ωi = ±1 and Respi ,qi ωj = 0 for any j 6= i,
and where ωi is the limit of ωi,t ∈ H 0 Ω1Xt . The extension
data in the LMHS arising from
W (N)
0 → W0 (N) → W1 (N) → Gr1
→0
is given by Σi AJXe0 (pi − qi )
In summary, Hodge theory plus heuristic geometric reasoning
guides in what to look for when we compactify Mg .
43/82
To complete the story from an algebro-geometric perspective,
once we are led to try to add nodal curves to complete Mg ,
the question arises: Which nodal curves should be added?
Hodge theory does not distinguish between
-
-
C
44/82
If we let X0 be the singular curve, then for g = 2 we may
impose the condition
ωX0 is ample.
π
e
e→
More precisely, to a family X
− ∆ where
e
I X is smooth and has no −1 curves in the fibres,
et is smooth for t 6= 0 and the central fiber X
e0 is nodal
I X
⊗k
we apply the relative pluricanonical mapping given by Rπe0 ωX/∆
e
for k = 3 to obtain
π
X→
− ∆,
45/82
e0 with
which has the effect of contracting all curves C ⊂ X
ωXe /∆ · C = 0,
these being the curves such as C above. These turn out to be
exactly the stable curves X0 with χ(OX0 ) = 1 − g .
46/82
Another way of describing this from an algebro-geometric
perspective is this: The minimal model program (MMP) for
surfaces (which is classical, dating to the Italian school)
applied to families X → ∆ gives the existence of an abstract
compactified moduli space Mg . Hodge theoretic
considerations as above then serve as a guide to what the
singular curves in the boundary ∂Mg = Mg \Mg should be.
47/82
As we shall explain below, for surfaces of general type the
existence of an M has been proved [Kollar, Shepherd-Barron,
Alexeev] using the MMP for threefolds, but except in some
simple cases what singular surfaces appear in ∂M is not
understood. It is here that ones hopes that Hodge theory may
provide some guidance.
<o>
48/82
Monodromy cones
Above we have taken A = I ; what about other A’s in the cone
A = t A, A > 0? For example, for g = 2 and for the
degeneration
δ3
δ3 =δ1 +δ2
δ2
=
=
A = ( 21 12 ) = ( 10 00 ) + ( 00 01 ) + ( 11 11 )
=
δ1
A1
A2
A3
49/82
A is in the interior of the cone spanned by A1 , A2 , A3 . What is
suggested is that we consider several parameter families of
algebraic varieties giving rise to a VHS
Φ : S → Γ\D
where S = S\Z with the sub variety Z corresponding to
singular varieties. By localizing around a point of Z we may
assume that S = ∆∗` and
Γ = {T1 , . . . , T` },
Ti = exp Ni
where Ni ∈ EndQ (V ) are commuting nilpotent elements.
50/82
Definition
A several variable nilpotent orbit is given by
X
exp
zi Ni · F0
where the Ni are commuting elements in EndQ (V ) and where
the conditions
P
(i) exp( i zi Ni ) · F0 ∈ D for Im zi 0,
(ii) Ni F0p ⊂ F0p−1
are satisfied.
51/82
Schmid’s theorem extends to the several variable case, so that
from the perspective of the asymptotics of VHS’s it suffices to
consider nilpotent orbits. There is also a several variable
sl2 -orbit theorem [Cattani-Kaplan-Schmid] and
[Kashiwara-Kawai], but it is rather involved to state and will
not be needed here.
52/82
Note
From an algebro-geometric perspective there are well-defined
cohomological obstructions to completing the Ni ’s to
commuting sl2,i ’s.
possible
not possible
This can be understood for curves but essentially nothing is
known in the general case.
53/82
The monodromy cone
σ = {Nλ =
X
λi Ni ,
λi > 0}
i
has remarkable properties, including
interior: W (Nλ ) is independent of Nλ ∈ σ,
faces: W (N) is a relative weight filtration for W (Ni ).
Nilpotent orbits have the property that LMHS associated to
(V , W (Nλ ), F0 ) is independent of the Nλ ∈ σ. Here they will
also have the property that each Nλ gives a PLMHS, where
the primitive pieces will in general depend on Nλ . The above
properties are not linear algebra results; they require deep use
of Hodge theory.
54/82
Example
For the VHS given by independently smoothing nodes in the
last example just above, taking
!
0 Ai
Ni =
0 0
gives a monodromy cone. This one is in fact maximal. The
larger the cone the more singular the varieties corresponding
to its interior is.
55/82
Thus
←→
σ = spanR>0 {N1 , N2 }
←→
σmax = spanR>0 {N1 , N2 , N3 }
?
and σ is a face of σmax
56/82
In general, weight one VHS’s have the special properties that
the weight filtrations
W1 = W0⊥
∗ ∗ ∗
correspond to maximal parabolic subalgebras 00 ∗0 ∗∗ of
EndQ (V ). For each g1 = dim W0
I they are all conjugate,
I their unipotent radicals are abelian.
This latter implies that there are maximal monodromy cones;
any σ is a face of a maximal one corresponding to all A’s with
A = t A, A > 0.
W0 ⊂ W1 ⊂ W2 = V ,
<o>
57/82
Returning to limiting mixed Hodge structures we have the
Definition
The type of a LMHS (V , W (N), F ) is the collection of Hodge
W (N)
numbers of the PHS’s Grn+k V prim .
58/82
Examples
r
n=1
r
r
r
r
r
N1
r
n=2
the type is uniquely determined
by rank N
N
r
rr
r N0 r
N0
r
N1
the type is uniquely determined
by rank N0 and rank N1
r
59/82
For n = 1 the conjugacy classes of nilpotent cones fall into
disjoint subsets indexed by one number, and within each type
there is a unique maximal monodromy cone of that type. This
is a reflection of the linearity in N of the polarizing condition
W (N)
Q(Nu, ū) > 0 for u ∈ Gr2
. We may picture this as
N1 ≺ N2 ≺ · · · ≺ Ng .
A face of a cone in Nk is in some N` for ` 5 k. This partial
ordering among the monodromy cones leads to a stratification
of ∂Mg .
60/82
For n = 2 with h2,0 = 2, Robles, Pearlstein and Brosnan have
classified the nilpotent cones. The story here is quite a bit
more subtle. First, as a reflection of the non-linearity in N of
the polarizing conditions
Q(N 2 u, ū) > 0,
W (N)
u ∈ Gr4
V
maximal nilpotent cones may not exist. Secondly, the strict
partial ordering of the monodromy cones of a given type we
saw in the n = 1 case may not hold. This has the
algebro-geometric implication that degenerations among
certain types of algebraic varieties in the boundary of a moduli
space may not be possible. Following is Robles table.
61/82
r
r
I
r
r
r
N 2 = 0, rank N = 2
r
N
r
r
r
r
II
N 3 = 0, rank N = 1
r
r
The notation in II means that N 2 6= 0.
62/82
r
N
III
r
r
r
N 2 = 0, rank N = 4
r
N
r
r
N1
IV
N0
r
r
r
N0
r
N03 = 0, N12 = 0
rank N0 = 1, rank N1 = 2
r
N1
63/82
r
N
r
V
N 3 = 0, rank N = 2
N
r
Her incidence table is
(
NI ≺
NII
NIII
)
≺ NIV ≺ NV .
Thus, type II cannot degenerate to type III.
64/82
<o>
We will now give an informal discussion of the question:
Can we use the above to say something about the
boundary of the moduli space of surfaces with
h2,0 (X ) = 2,
and in the last part of this talk will give the first part of an
example in response to this question.
65/82
For surfaces of general type the KSBA (Kollár,
Shepherd-Barron, Alexeev) compactification M of the moduli
space M is obtained by applying the MMP for threefolds to
degenerations
X→∆
to see what type of singular surfaces X0 should be added to
the smooth ones.1 Using the general theory from the MMP,
this compactified moduli space M may be proved to exist as a
complete, separated algebraic space (a somewhat more general
notion than an algebraic variety, but it’s OK for present
purposes to think variety).
1
More precisely, to surfaces X that are smooth except for rational
double points (RDP’s), these being the ones that are contracted under all
the pluricanonical maps ϕmKX given by |mKX |. The singularities of the
surfaces that are added are essentially
(i) a double curve with pinch points (no triple points);
(ii) a few isolated singularities, including simple elliptic ones and cusps.
66/82
As note above, it seems that there are essentially no
non-classical (i.e., whose period domain is not an HSD)
examples where the limit surfaces X0 ∈ ∂M needed to
compactify the moduli space have been described.
If one were to use Hodge theory as a guide to which X0 ’s to
add, one might envision three steps:
(A) Use a known stratification of the possible monodromy
cones to suggest a stratification of ∂M;
(B) Contruct algebro-geometric degenerations of surfaces that
realize those suggested by (A);
(C) For each stratum in (B), define the “type” of singular
surfaces and show that any surface of that type occurs in
that stratum in M.
67/82
For curves, the type means the arithmetic genus χ(OX0 ) and
the dual graph of X0 , where we assume that X0 is nodal and
that ωXe0 (log D) is ample.
Example
An H-surface is a minimal algebraic surface X that
I is of general type;
I has h 2,0 (X ) = 2 and h 1,0 (X ) = 0 (regularity);
I has K 2 = 2.
X
68/82
Then X has only RDP’s, and if it is smooth, the
Riemann-Roch theorem gives
1
χ(OX ) =
KX2 + χtop (X )
12
w

b2 (X ) = 34,
h1,1 (X ) = 30.
The above H stands for Horikawa, who analyzed in detail the
smooth H-surfaces via their bi-canonical map
ϕ2K : X _ _ _/ P4 .
The image of this map is given by the equations at the
beginning of this talk. We shall use Hodge theory as a guide
to construct a degeneration of type I; thus for H-surfaces one
has (A) in general and partial results for (B).
69/82
Hodge theory will be used in two ways:
(i) to suggest what to look for, and
(ii) to suggest where to look.
For degenerations of type I, the simplest possibility is that as
X → X0
I X0 acquires an elliptic double curve D;
I one ω ∈ H 0 (Ω2 ) tends to ω0 ∈ H 0 (Ω2 ).
X
e
X
0
Regarding the second point, as X → X0 one expects all of the
2-dimensional vector spaces H 0 (Ω2X ) to go in the limit to
e
e0 ) = 1, the ω0
H 0 (Ω2Xe (log D)).
Since we should have h2,0 (X
0
above remains holomorphic in the limit and its divisor
(ω) → D, while a general ψ ∈ H 0 (Ω2X ) tends to ψ0 with a
log-pole on D and whose residue generates H 0 (Ω1De ).
70/82
So where is the double curve D going to come from? The
canonical series |KX | is a pencil of curves C . Horikawa shows
that for a general X
I |KX | has no fixed component,
I the pencil |KX | has K 2 = 2 base points.
X
71/82
Blowing up the two base points we get X̂ with the picture
Ct
C∞
X̂
E1
E12 =E12 =−1
t
E2
?
P1
∞
where a smooth C ∈ |KX | has genus
1
g (C ) = (KX2 + KX · C ) + 1 = 3.
2
We note that
KX̂ = (C ) + 2(E1 + E2 ).
72/82
A nice little exercise is to compute
(i) # nodal curves δ + 1 = 42
(ii) # hyperelliptic fibres = 1
For (i), each node adds +1 to χtop (P1 ) · χtop (C ), which would
be the Euler characteristic of X̂ if there were no singular fibres.
For (ii), the argument is more subtle. There is a linear relation
among λ = degree (Hodge bundle), δ1 , and δ0 = #
hyperelliptic curves in the fires of X̂ → P1 . Since δ0 is not
detectable Hodge-theoretically, this is insufficient to give (ii).
73/82
Lemma
Rf0 ωX̂ /P1 ∼
= OP1 (1) ⊕ OP1 (1) ⊕ OP1 (3).
Proof
3
It is known from Hodge theory that Rf0 ωX̂ /P1 = ⊕ OP1 (ki )
i=1
where
I all ki = 0
(non-negativity of the Hodge bundles);
I in fact, all ki > 0
since q(X ) = 0.
0
0
0
2 ∼
0
2
∼
I H (R ω
f X̂ /P1 (−2)) = H (ΩX̂ ) = H (ΩX ).
This map uses OP1 (1) ∼
= OP1 (∞) and is given by
ϕ(t) −→ ϕ(t) ∧ dt = Φ.
74/82
Since h0 (Ω2X ) = 2 it follows that k1 + k2 + k3 = 5, which gives
the possibilities
(
(1, 2, 2)
(1, 1, 3).
If the first of these holds, then we would have a
ϕ ∈ H 0 (Rf0 ωX̂ /P1 (−2)) which doesn’t vanish on any fibre
which implies that KX = OX .
From this we find that λ = deg Rf0 ωX̂ /P1 = 5.
75/82
To find what D should be we imagine that as X → X0 the
section ϕ(t) which vanishes to 2nd order at t = 0 tends to ϕ0
where
lim(ϕ(t) ∧ dt) = C0,∞ .
Thus we may hope to have on X̂0
D = C0,∞ + 2(E1 + E2 ).
C0,∞ is a curve with χ(OC0,∞ ) = −4 with nodes at the
e0,∞ is an elliptic curve. Moreover, the
Ei · C0,∞ . Thus C
differential ϕ0 ∧ dt on X0 has no zeroes. Thus we may
suspect, and it may be proved, that
X0 is a polarized K3 surface.
76/82
Note
The above argument gives
(Φ) = (ϕ) + 2(E1 + E2 )
where (ϕ) is the divisor of
ϕ ∈ H 0 (OP1 (1)) ∼
= H 0 (Rf0 ωX̂ /P2 (−2)). The intersections
(E1 + E2 ) · Ct are bi-tangents to the place quartics ϕKCt (Ct ).
77/82
It remains to actually write down the equations of a family of
surfaces that actually realize the above “thought example.” In
the weighted projective space P(1, 1, 2, 2) with homogeneous
coordinates [t0 , t1 , x3 , x4 ] we consider the surface given by
(∗)
t02 G (t02 , t0 t1 , t12 , x3 , x4 ) = F (t02 , t0 t1 , t12 , x3 , x4 )2
where F (x), G (x) are general homogeneous polynomials of
degrees 2, 3 in x = [x0 , x1 , x2 , x3 , x4 ]. The H-surface is the
desingularization X of the surface (∗).
78/82
Geometricaly we are embedding

/ P4
∈
∈
P(1, 1, 2, 2) (t0 , t1 , x3 , x4 )
/
[t02 , t0 t1 , t12 , x3 , x4 ].
The image is the singular quadric x0 x2 − x12 = 0 in P4 .
Blowing up P4 along x0 = x1 = x2 = 0 gives a
g 2, 2) of P(1, 1, 2, 2). In fact,
desingularization P(1,1,
e 1, 2, 2) = P (OP1 ⊕ OP1 ⊕ OP1 (2)) ∼
P(1,
= R(Rf0 ωX̂ /P1 )
∈
∈
and the proper transform of the surface (∗) is the surface X̂
over P1 :
P(1, 1, 2, 2) _ _ _ _/ P1
[t0 , t1 , x3 , x4 ]
/
[t0 , t1 ].
79/82
The surface (∗) in P4 is the bi-canonical model of X . It is
singular along the plane conic
x0 = x1 = F (x) = 0,
which is a double curve with eight pinch points. If we now
consider the surface
(∗ ∗)
t02 t12 Q(x) = F (x)2
where Q(x) is a general quadric, then we obtain a surface
which we may think of as the degeneration of (∗).
80/82
This provides a geometric realization of the thought example
above. Thus in conclusion, we have
I
Hodge theory _ _ _/ thought example
I
thought example _ _ _/ actual example.
Although promising, and there is a very nice interplay between
Hodge theory and the very beautiful algebraic geometry of the
H-surface and its limit, it is only a first step.
<o>
81/82
Recent Developments
t02 G = F 2
0
?
t02 t12 Q = F 2
I
Q
Q
QQ
s
+
? II
Q
Q
Q
s
Q
III
t02 (t0 − t1 )2 t12 L = F 2
(t0 R + F )(t0 R − F ) = 0 (∗)
+
IV in (∗) allow F = R = 0 to acquire a node
?
V allow F = R = 0 to acquire a second node
Hodge theory suggests two algebro-geometric components for
type III degenerations, and these illustrate two such
possibilities.
82/82