Pure Motives and Rigid Local Systems
Spring 2014, taught by Stefan Patrikis.
Contents
1 Motivations
1.1
2
Towards a Motivic Galois Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Weil Cohomology
2.1
4
4
The Trace in Betti Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
3 Algebraic de Rham Cohomology
7
4 Cohomological Correspondences
14
5 Intersection of Cycles
18
6 Adequate Equivalence Relations
20
7 Tannakian Theory
24
8 Standard Conjectures
26
8.1
The Künneth Conjecture C(X) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
8.2
The Lefschetz Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
8.3
The Hodge Standard Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
8.4
Dependencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
9 Motivated Cycles
34
10 Source of Motivated Cycles
39
11 Rigid Local Systems
46
12 Perverse Sheaves
54
13 Middle Convolution
56
14 Construction of G2 Local Systems
62
1
15 Universal Rigid Local Systems
66
15.1 Middle Convolution With Parameters
. . . . . . . . . . . . . . . . . . . . . . . . . .
67
16 Motivic Nature
70
17 Alternate Methods
74
1
Motivations
Let k be a field (for simplicity, of characteristic zero, with an embedding of k into C). We consider smooth projective varieties over k. Given such a variety X, we may consider three forms of
cohomology:
∗ (X) = H ∗ (X an , Q), a Q-vector space.
1. The Betti cohomology HB
sing
C
∗ (X/k) = H∗ (X, Ω•
2. The (algebraic) de Rham cohomology HdR
X/k ), a k-vector space.
∗ (X , Q ), a Q -vector space.
3. The `-adic cohomology H`∗ (X) = Hét
`
`
k
Theorem 1.1 (Comparison Isomorphism). There exist functorial isomorphisms
∼
∗
∗
αB,dR,X : HB
(X/k) ⊗k C.
(X) ⊗Q C −
→ HdR
(1.1)
Functoriality means that given f : X → Y , the diagram
∗ (X) ⊗ C
HB
αX
∗
fB
∗ (Y ) ⊗ C
HB
∗ (X/k) ⊗ C
HdR
∗
fdR
αY
(1.2)
∗ (Y ) ⊗ C
HdR
commutes (as well as some other properties).
∗ and im f ∗ ∼ im f ∗ .
Here are some consequences: After tensoring with C, ker fB∗ ∼
= ker fdR
B =
dR
This goes along with the slogan that “sufficiently geometric” pieces of cohomology have comparable
meaning in every H ∗ . This suggests that there should be an abelian category in the background.
Furthermore, a standard conjecture (due to Künneth) suggests αX preserves degrees, so this category
would be graded.
Another standard conjecture (due to Lefschetz): Take an ample line bundle L on X. We obtain
2
dR ∈ H 2 (X)(1). Then α (η B ) = η dR , and α is compatible
classes η B = cB
X
X
1 (L) ∈ HB (X)(1) and η
dR
with cup product. This means that for i ≤ dim X = d, the primitive cohomology
i
PrimiηB (X) = ker η d−i+1 : HB
(X) → H 2d−i+2 (X)
2
(1.3)
L
which yields a decomposition H ∗ (X) =
Primi ∪ η ∗ , PrimiηB (X) gets mapped to PrimiηdR (X)
after tensoring with C. This should imply than Primitive cohomology should be “sufficiently geometric”.
Extrapolating, we get a powerful heuristic for transferring properties between Betti, `-adic, and
de Rham cohomology.
By the early 60’s, one knew (or conjectured):
k (X) naturally carries a pure Hodge
1. Hodge theory: for X/C a smooth projective variety, HB
L
p,q such
structure of weight k. This is a Q-vector space V and a bigrading VC =
p+q=k V
that V p,q = V q,p (the bar being complex conjugation with respect to VR ).
2. It was conjectured that for smooth projective varieties X/Fq , H`k (X) Gal(Fq /Fq ) is pure of
weight k. This means that for f rq the geometric Frobenius, all eigenvalues of f rq are algebraic
k
numbers whose absolute value under every embedding into C equals q 2 .
As an example, H`2 (P1 ) is the inverse of the `-adic cyclotomic character ω` . We denote this
Galois representation by Q` (−1).
In the non-pure case, there will be a weight filtration. An example is to take X a smooth
projective curve, S a finite set of points, and U = X \ S. We have an exact sequence
0 → H 1 (X) → H 1 (U ) →
H 2 (X)
| S{z }
→ H 2 (X) → 0.
(1.4)
supported on S
H 1 (X) is pure of weight 1 while H 2 (X) = Q` (−1) is pure of weight 2. Meanwhile HS2 (X) =
H 0 (S)(−1) = Q` (−1)#S is also pure of weight 2. Hence H 1 (U ) has an increasing weight filtration:
if Wi is the weight ≤ i piece, W0 = 0, W1 = H 1 (X), and W2 = H 1 (U ).
∗ (X) (for X not necessarily smooth and projective) a mixed Hodge structure, the key
To give HB
∗ such that the E p,q term is (conjecturally) pure of
point is to find a spectral sequence E =⇒ HB
2
weight p + 2q.
∗ (U ) a mixed Hodge structure for nonsingular U (no longer proDeligne in Hodge II gives HB
jective). The `-adic analogue uses the Leray spectral sequence for jS: U ,→ X, for X the smooth
completion. We’ll assume that X \ U is a union of smooth divisors i∈I Di with normal crossings.
We have
E2p,q = H p (Xk , Rjq∗ Q` ) =⇒ H p+q (Uk , Q` ).
(1.5)
Fact.
Rjq∗ Q` =
M
Q` (−q)DQ =Si∈Q Di .
(1.6)
H p (DQ , Q` )(−q)
(1.7)
Q⊆I
|Q|=q
This implies
E2p,q =
M
|Q|=q
3
which (from the Weil conjectures) is pure of weight p + 2q. As the Erp,q are subquotients of E2p,q ,
they are also pure of weight p + 2q.
Now d3 : E3p,q → E3p+3,q−2 maps from a space of weight p + 2q to one of weight p + 2q − 1, and
the eigenvalue mismatch forces d3 = 0. Similarly dr = 0 for r ≥ 3. This implies Leray degenerates
at E3p,q . We have

E3p,q = ker 
M
H p (DQ , Q` )(−q) →
|Q|=q
M
,
H p+2 (DQ0 , Q` )(1 − q) im(· · · )
(1.8)
|Q0 |=q−1
where the above morphisms are morphisms of Galois modules. We want the Betti analogues
to take maps of pure Hodge structures. This is proven by the reinterpretation: if Q0 ⊆ Q, then
DQ ,→ DQ0 and the dp,q
2 ’s are simply Gysin maps (Poincaré dual to pullback), which are indeed
maps of pure Hodge structures.
In summary, `-adic Leray gives a weight filtration (which is the Leray filtration up to shift)
whose graded pieces are
grnW H k (U ) = E32k−n,n−k
(1.9)
k (U ), such that we already
pure of weight n. The Betti Leray also gives a “weight” filtration on HB
p,q
know that the E∞ are naturally pure Hodge structures.
1.1
Towards a Motivic Galois Formalism
In classical Galois theory, let k be a field and k s a separable closure. Then there is an equivalence
between finite étale k-schemes and finite sets with a continuous Γk = Gal(k s /k)-action, by mapping
Z to Z(k s ) (viewed as a Galois module). Linearizing this equivalence, we get a map from artin
motives to finite dimensional Q-vector spaces with continuous Γk -action. Here artin motives are
(numerical) motives built out of zero-dimensional varieties. The linearized map is taking H 0 .
We aim to find a higher dimensional version. The set of artin motives is contained within the
“pure homological motives”, which by standard conjectures should be mapped to representations of
some group Gk . This views “motivic Galois theory” as an extension of classical Galois theory: there
will be an exact sequence
1 → Gk → Gk → Γk → 1.
(1.10)
An application of rigid local systems is to construct the exceptional group G2 as a quotient of
GQ .
2
Weil Cohomology
We restrict our attention to smooth projective varieties X over a field k. Let P(k) be the category
of these varieties. P(k) is a symmetric monoidal category via the fiber product, having the obvious
associativity and commutativity constraints, and the unit being Spec k.
4
Another example of a symmetric monoidal category (or ⊗-category): Let E be a field, and
be the category of finite dimensional graded E-vector paces in nonnegative degrees. Here the
operation is the usual tensor product, where the grading is given by
Gr≥0
E
(V ⊗ W )n =
M
V i ⊗ W j.
(2.1)
i+j=n
∼
We give this category the graded commutativity constraint; that is, maps CV,W : V ⊗ W −
→
W ⊗ V by
v ⊗ w 7→ (−1)deg(v) deg(w) w ⊗ v.
(2.2)
A hWeil cohomology over a field E of characteristic zero on P(k) is a ⊗-functor H ∗ : P(k)op →
Gr≥0
H ∗ being a ⊗-functor means that it comes with functorial (“Künneth”) isomorphisms KX,Y :
E
i
∼
H ∗ (X) ⊗ H ∗ (Y ) −
→ H ∗ (X × Y ) respecting graded commutativity , satisfying:
1. (Normalization) dimE H 2 (P1 ) = 1. In particular, H 2 (P1 ) is invertible in GrE , so we may define
“Tate twists” in GrE by taking
V (r) = V ⊗ H 2 (P1 )⊗(−r) .
(2.3)
2. (Trace axiom) For every X of dimension d, there is a linear map trX : H 2d (X)(d) → E
satisfying:
(a) Under the Künneth isomorphisms, trX×Y = trX · trY .
(b) trX and “cup product” induce perfect dualities
tr
H i (X) × H 2d−i (X)(d) → H 2d (X)(d) −−X
→ E.
(2.4)
3. (Cycle class maps) Let Z r (X)Q be the Q-vector space having a basis consisting of the integral
r : Z r (X) → H 2r (X)(r)
closed subschemes Z ,→ X of codimension r. Then there exist maps γX
satisfying:
r factor through rational equivalence, meaning they factor through the Chow groups
(a) γX
CH r (X).
(b) The maps satisfy contravariant functoriality in X. That is, given
X
f
Y
(2.5)
−1 f Z
Z
we have
r
f ∗ γYr (Z) = γX
([f −1 Z])
f∗ γX = γY f∗ .
5
(2.6)
(2.7)
when this makes sense. This will always make sense after passing to CH r . In general,
we can’t always define f ∗ on Z r (Y ), but we can if f is flat, for then f −1 Z has all of its
irreducible components of codimension r in X. To f −1 Z, we associate the cycle
[f −1 Z] =
X
ni Wi
(2.8)
where the Wi are the irreducible components made reduced and ni = length(Of −1 Z,Wi ).
We’ll later define f∗ .
(c) For α ∈ Z r (X) and β ∈ Z s (Y ),
r+s
r
γX
(α) ⊗ γYs (β) = γX×Y
(α × β).
(2.9)
(d) (this pins down the trace) The composite is described by
Z d (x)
P
ni Pi
d
γX
trX
H 2d (X)(d)
P
E
(2.10)
ni [k(Pi ) : k]
if the Pi are closed points.
Here is the source of the “cup product”: let ∆ : X ,→ X × X be the diagonal embedding. We
then have
H ∗ (X × X)
H ∗ (X)
^
KX,X
(2.11)
H ∗ (X) ⊗ H ∗ (X)
2 (P1 ) =
Remark. We set HB
1
2πi Q.
This means that via the comparison isomorphism
∼
2
2
(P1 /Q) ⊗Q C
HB
(P1 ) ⊗Q C −
→ HdR
(2.12)
2 (Q) is 1 H 2 (P1 /Q). This hints at the theory of “periods”: take for granted
the image of HB
2πi dR
that αB,dR is compatible with Mayer-Vietoris. Write P1 = (P1 \ 0) ∪ (P1 \ ∞), and then we have,
after tensoring with C,
0
1 (P1 \ 0, ∞)
HB
2 (P1 )
HB
∼
0
0
(2.13)
∼
1 (P1 \ 0, ∞)
HdR
2 (P1 )
HdR
0
Do the calculation for the H 1 ’s. The inverse of the map takes
Z ω→
7
σ 7→
ω .
σ
6
(2.14)
1 (P1 \ 0, ∞) is given by dz , which is mapped to a function taking a counterA Q-basis for HdR
z
1 (P1 \ 0, ∞) → 1 H 1 (P1 \ 0, ∞).
clockwise loop to 2πi. Hence HB
2πi dR
2.1
The Trace in Betti Cohomology
Let X/C be a smooth projective variety of pure dimension d. Define trX as the composite
2d
HB
(X)(d)
1
(2πi)d
α
2d
−−−−→ HB
(X) −
→ H 2d (X, Q)
(2.15)
∼
(the last cohomology being sheaf cohomology) and
R
β
H 2d (X, Q) ⊗ R −
→ H 2d (Γ(X, A•X )) −−X
→ R.
∼
(2.16)
Here A•X is the C∞ de Rham complex with real coefficients. Then check that the image of the
composite is actually contained in Q.
α is a generality for every locally contractible, locally path-connected topological space. (The
sheafy singular cochain complex is a flasque resolution of Q.) β is the Poincaré lemma (A•X is a fine
resolution of R).
3
Algebraic de Rham Cohomology
∗ (X/k) = H∗ (X, Ω•
Let k be a field of characteristic zero and X/k smooth. Define HdR
X/k ).
∗ : P(k)op → Gr≥0 is a Weil cohomology.
Theorem 3.1. HdR
k
We’ll selectively check some axioms.
∗ is a ⊗-functor.
Lemma 3.2. HdR
Proof. Given f : X → Y , we obtain a pullback map f ∗ : H∗ (Y, Ω•Y ) → H∗ (X, Ω•X ), given by pulling
back forms. That is, given an injective resolution Ω•Y → I • , f −1 Ω•Y → f −1 I • is a quasi-isomorphism,
and then we can choose an injective resolution J • of f −1 I • . The maps on global sections then give
maps
H∗ (Y, Ω•Y ) → H∗ (X, f −1 Ω•Y ) → H∗ (X, Ω•X ).
(3.1)
The ⊗-structure is obtained as follows: given bounded below complexes F • and G • on X, we
get a map
H∗ (X, F • ) × H∗ (X, G • ) → H∗ (X, F • ⊗ G • ).
(3.2)
∧
In the case X = Y , composing with the exterior product Ω•X × Ω•X −
→ Ω•X gives the cup product.
In general, we obtain the Künneth isomorphisms.
∗ (X/k) is finite dimensional for X/k smooth and projective. We
We also need to check that HdR
have the Hodge-de Rham spectral sequence
7
∗
E1p,q = H q (X, Ωp ) =⇒ HdR
(X)
(3.3)
and the E1p,q are finite dimensional, and zero unless p, q are both nonnegative.
Next, we consider the normalization axiom. To compute algebraic de Rham cohomology, we split
∗ (U ) = H ∗ (Γ(U, Ω• )). For example, for A1 ,
a general variety up into affines. If U is affine, then HdR
U
d
0 (A1 ) = k and H 1 (A1 ) = 0.
our complex is k[T ] −
→ k[T ] dT and so HdR
dR
Our assertion on de Rham cohomology for affine U is true by vanishing of H i (U, F) for F
quasicoherent and i > 0.
∗ from Čech resolutions. Let U = {U }
More generally, we can compute HdR
i i∈I be an open affine
cover. Then for any quasicoherent sheaf F on X, we get an acyclic Čech resolution
∼
F−
→ Cˇ• (U, F)
(3.4)
where
Cˇq (U, F)(U ) =
M
Γ(U ∩ UJ , F)
(3.5)
|J|=q+1
UJ =
\
Uj .
(3.6)
j∈J
∗ , we have the double complex
For HdR
Cˇ• (U, OX )
Cˇ• (U, Ω1X )
···
(3.7)
Ω1X
OX
···
∗ (X/k) as its cohomology. Recall that the
and the global sections of its total complex gives HdR
••
total complex of a double complex D is
Totn (D•• ) =
M
Dp,q .
(3.8)
p+q=n
We can use this to verify the normalization axiom. We express P1 as the union
P1 = (P1 \ ∞) ∪ (P1 \ 0) = U0 ∪ U1 .
(3.9)
The double complex is zero outside of the region
Ω1 (U0 ∩ U1 )
O(U0 ∩ U1 )
(3.10)
Ω1 (U0 ) ⊕ Ω1 (U1 )
O(U0 ) ⊕ O(U1 )
8
so the total complex is given by
O(U0 ) ⊕ O(U1 )
(a0 , a1 )
Ω1 (U0 ) ⊕ Ω1 (U1 ) ⊕ O(U0 ∩ U1 )
h
i
(da0 , da1 ), −a0 + a1
Ω1 (U0 ∩ U1 )
2 is 1-dimensional, generated by
It is a simple calculation to show that H 1 = 0 while HdR
(3.11)
dt
t .
This is really saying
0
HdR
(P1 ) = H 0 (P1 , OP1 )
(3.12)
2
HdR
(P1 ) = H 1 (P1 , Ω1P1 ).
(3.13)
A generalization of this is:
Proposition 3.3 (Hodge-de Rham spectral sequence). There is a spectral sequence
p+q
E1p,q = H q (X, ΩpX ) =⇒ HdR
(X).
(3.14)
∗ (X) is the Hodge filtration.
The induced filtration on HdR
The filtration arises from filtering the Čech double complex. Take
F p Tot• (D•• ) =
M
Da,•−a .
(3.15)
a≥p
In particular, the pth graded piece is Dp,•−p .
In general, a filtered complex gives rise to a spectral sequence
E1p,q = H p+q (grp Tot• )
p,q
E∞
p
= gr (H
p+q
•
Tot )
(3.16)
(3.17)
where the filtration on H ∗ (Tot• ) is defined by
h
i
im H n (Filp Tot• ) → H n (Tot• ) .
(3.18)
In our situation,
E1p,q = H p+q (Dp,•−p )
ker(Dp,q
Dp,q+1 )
→
→ Dp,q )
= H q (X, Ωp )
=
im(Dp,q−1
where the last equality follows from Čech theory.
9
(3.19)
(3.20)
(3.21)
Remark. If we filtered the other way,
(F 0 )p Tot• =
M
D•−a,a
(3.22)
p
HdR
(UJ ).
(3.23)
a≥p
then we would have
(E10 )p,q =
M
|J|=q+1
This gives us the Mayer-Vietoris spectral sequence.
Theorem 3.4. If char(k) = 0, then the Hodge-de Rham spectral sequence degenerates at E1p,q .
We’ll sketch the remaining axioms. We seek a trace map giving Poincaré duality, as well as cycle
class maps.
The trace map can be obtained from the following two steps:
2d (X) is abstractly isomorphic to H 0 (X, O ) (which is k is X is geometrically
1. Show that HdR
X
connected). This will follow from Serre duality.
2. Pin down the actual trace map. This will happen after constructing the cycle class maps.
Recall from Serre duality, we have the left map in
Hd (X, Ω•X [d])
H d (X, ωX )
(3.24)
H2d (X, Ω•X )
k
2d (X) is the map E d,d →
where ωX = ΩdX is the dualizing sheaf. The map H d (X, ΩdX ) → HdR
1
2d (X). We want to show it’s an isomorphism. Since the domain is free of rank 1 over H 0 (X, O )
HdR
X
d−1,d
0
is zero, and enough
and the map is a map of H (X, OX )-modules, it will suffice to show that d1
2d (X) is nonzero. (We won’t assume degeneration of the spectral sequence to show
to show that HdR
this.)
This can be checked by reducing to the case of X = Pd . Choose a finite flat map to some
projective space, and then there is a trace map
X
∗ (X)
HdR
f
Pd
trf
(3.25)
∗ (Pd )
HdR
such that trf ◦f ∗ = deg f . (We know this at the level of rings of functions.) Then f ∗ : H 2d (Pd ) →
2d (Pd ) 6= 0. But now a direct calculation of the
is injective, so it’s enough to show that HdR
Hodge-de Rham spectral sequence shows this vector space is isomorphic to H d (Pd , ΩdPd ) ∼
= k.
2d (X)
HdR
10
2d (X) and then set, for c ∈ H 0 (X, O ,
Now, we will carefully choose a generator uX of HdR
X
trX (c · ux ) =
trH 0 (X,OX )/k (c)
[H 0 (X, OX ) : k]
(3.26)
In other words, trX (uX ) = 1.
To define uX , let P be a closed point of X. If ch is the Chern character, then
chdR (κ(P )) = [k(P ) : k]uX
(3.27)
2d (X) independently of P . Here κ(P ) is a skyscraper sheaf at P .
will define uX ∈ HdR
d (P ) = [k(P ) : k]. Here, γ d (P ) will be
Recall that we want the cycle classes γX to satisfy trX γX
X
defined to be chdR (κ(P )), so that
d
trX γX
(P ) = trX (chdR (κ(P )))
= [k(P ) : k].
(3.28)
(3.29)
We’ll show how to construct all of the cycle classes:
1. Define the first Chern class c1 (L) for line bundles L.
2. Define Chern classes cdR (E) for vector bundles E.
3. Produce the Chern character chdR from cdR .
4. Know that we have a factorization
VectX
chdR
L
H 2i (X)
(3.30)
K 0 (X)
where K 0 (X) is the Grothendieck group of VectX . For X smooth. it’s isomorphic to the
Grothendieck group K0 (X) of coherent sheaves on X.
p
5. For Z ,→ X a codimension p cycle, we define γX
(Z) = chdR
p (OZ ).
Now we’ll verify these steps:
×
2 (X)(1) (recall that H 1 (X, O × ) = Pic X).
1. We want a group homomorphism H 1 (X, OX
) → HdR
X
This is the map induced from
OX
Ω1X
d log
×
OX
11
Ω2 X
(3.31)
2. Given a vector bundle E of rank r on X, form the projective bundle P(E), which we define
to be the bundle of lines in E. Given the map f : P(E) → X, f ∗ E has the tautological line
2
subbundle L∨
E whose fiber at α 7→ x is the line α ⊆ Ex . Let c = c1 (LE ) ∈ H (PE). Then:
∗ (PE) is a free module over H ∗ (X) with basis 1, c, c2 , . . . , cr−1 .
Fact. HdR
dR
Then define cdR
j (E) by the formula
cr +
r
X
r−j
cdR
= 0.
j (E) ^ c
(3.32)
j=1
This agrees with the old definition of c1 (L), and is functorial in X.
Here is a key property: Define the total Chern class
cdR (E) =
r
X
cdR
j (E).
(3.33)
j=0
Then given a short exact sequence of vector bundles
0 → E 0 → E → E 00 → 0
(3.34)
we have c(E) = c(E 0 ) ^ c(E 00 ). A sketch of a proof goes as follows:
(a) Show that if E =
L
Li , then
Y
(1 + c1 (Li )).
c(E) =
(3.35)
Proof. After tensoring E with some
bundle, we many assume that each Li is very
Q line
d
∗
i
ample, giving us a map f : X → P with
Q Ldi i = fi (OPdi (1)), where di are as large as
we like. So we can reduce to the case X = P and Li is a component O(1).
Now each Li ⊆ E gives a section si of PE → P(Li ) with s∗i (L∨
E ) = Li . Pulling back, via
each si , the relation
cr +
X
cj (E) ^ cr−j = 0
(3.36)
we get, for xi = c1 (Li ),
(−xi )r +
X
cj (E)(−xi )r−j = 0.
(3.37)
So the polynomial
(−t)r +
X
∗
cj (E)(−t)r−j ∈ HdR
(X)[t]
(3.38)
∗ (X) ∼ k[x , . . . , x ]/(xdi +1 ), which forces
has x1 , . . . , xr as roots. As di 0, we have HdR
=
1
r
i
tr +
X
cj (E)tr−j =
Y
(t + xi )
so that cj (E) is the jth symmetric polynomial in the xi .
12
(3.39)
(b) (Splitting Principle) Reduce to this case by showing that there is some map f : Y →
X such that f ∗ (E, E 0 , E 00 ) split as direct sums of line bundles, and f ∗ is injective on
cohomology.
Proof. This proceeds in two steps:
i. Arrange for a pullback g : Y → X such that g ∗ E has a full flag of subbundles. This
is done by iterating the projective bundle construction.
ii. Split the extensions in the flag by a further pullback. Given a surjection E E 0 of
vector bundles on X, the space of sections of E → E 0 is an affine bundle over X.
Pulling back from X then splits the surjection, and because the bundle is affine, it
induces an isomorphism on cohomology.
3. Now we formally define the “Chern roots” of E as
c(E) =
r
Y
(1 + xi ).
(3.40)
i=1
We may consider any symmetric polynomial in the xi as a cohomology class. Then define the
Chern character of E by
ch(E) =
r
X
exi .
(3.41)
i=1
This character is additive in short exact sequences, and multiplicative in tensor products.
4. We get a ring homomorphism K 0 (X) → H 2• (X)(•). For X smooth, we can form finite locally
free resolutions of any coherent sheaf, producing an inverse to the natural map K 0 (X) →
K0 (X). Thus we get a map K0 (X) → H 2• (X)(•) which we will also denote by ch.
5. Finally, for Z ,→ X a cycle of codimension p, define the cycle class
p
(Z) = chp (OZ ).
γX
(3.42)
2d (X) (with d = dim X) is, for any closed point P of X,
In particular, our choice of basis for HdR
γX (P )
[k(P ) : k]
chd (κ(P ))
=
.
[k(P ) : k]
ux =
(3.43)
(3.44)
For the trace map, we still need to know:
• uX is independent of P . This follows roughly from invariance of ch in flat families, using a
curve to connect any two points.
13
• uX 6= 0. This follows from a reduction to projective space.
Then if P ∈ Pd (k), realize P inside a chain of projective spaces
P ,→ P1 ,→ P2 ,→ · · · ,→ Pd
(3.45)
and then, for a choice of section of OPn (1) for each n, use
0 → OPn (−1) → OPn → OPn−1 → 0.
(3.46)
So in K0 , we obtain
κ(P ) =
d
X
i=0
d
(−1)
OPd (−i).
i
i
(3.47)
Letting c1 (OPd (1)) = x and applying ch, we get
ch(κ(P )) =
X
d −ix
(−1)
e
i
i
(3.48)
= (1 − e−x )d
(3.49)
d
(3.50)
= (x + · · · )
≡ xd
(mod xd+1 )
(3.51)
We have xd+1 = 0, but xd 6= 0.
4
Cohomological Correspondences
Fix a Weil cohomology H ∗ : P(k)op → Gr≥0
E . Then given f : X → Y , define the Gysin map
∗
•−2(d
−d
)
X
Y
f∗ : H (X) → H
(Y )(−dX +dY ) given by the transpose of f ∗ after identifying cohomology
with the dual of homology under Poincaré duality. As a consequence, we have the projection formula
f∗ (f ∗ α ^ β) = α ^ f∗ β
(4.1)
trY (f∗ α ^ β) = trX (α ^ f ∗ β).
(4.2)
because
This also gives an alternative construction of cycle class maps (after knowing the trace exists):
i
for a smooth codimension p cycle Z ,−
→ X, we may define
p
γX
(Z) = i∗ ([Z])
(4.3)
where [Z] is the identity element of H 0 (Z). This can be extended to nonsmooth cycles by taking
a resolution
14
e
Z
(4.4)
τ
Z
i
X
p
e
and then defining γX
(Z) = τ∗ [Z].
A cohomological correspondence from X to Y is an element u ∈ H ∗ (X ×Y ). u may be interpreted
u∗
via Künneth as a linear map H ∗ (X) −→
H ∗ (Y ), where if u = a ⊗ b, then
u∗ (c) = trX (c ^ a)b.
(4.5)
Here the trace map is extended by zero away from the top degree. Another way of writing this
is
u∗ (c) = q∗ (p∗ (c) ^ u).
(4.6)
where p and q are the projections from X × Y to X and Y .
The element u also defines a map u∗ : H ∗ (Y ) → H ∗ (X) by b 7→ p∗ (u ^ q ∗ b).
The transpose of u is the image of u in H ∗ (Y × X) under swapping. Then u∗ = (t u)∗ .
We may compose correspondences. Suppose we are given u ∈ H ∗ (X × Y ) and v ∈ H ∗ (Y × Z)
Then define
v ◦ u = pXZ∗ (p∗XY (u) ^ p∗Y Z (v)) ∈ H ∗ (X × Z).
(4.7)
Lemma 4.1. (v ◦ u)∗ = v∗ ◦ u∗ .
Here are some ingredients to prove this:
• u∗ (a) ∈ H ∗ (X) is u ◦ a ∈ H ∗ (Spec k × X). Hence v∗ (u∗ (a)) = v ◦ (u ◦ a).
• The lemma then follows from associativity of composition of correspondences, which involves
manipulation of projection formulas. (See Chapter 16 of Fulton’s Intersection Theory.)
Also, if Γf is the cohomology class of the graph of a map f , we have (Γf )∗ = f ∗ and (Γf )∗ = f∗ .
So (t Γf )∗ = f ∗ . Everything here makes sense in Chow groups.
Proposition 4.2 (Lefschetz fixed-point theorem). Let k be algebraically closed and H ∗ a Weil
cohomology on P(k). Let X, Y ∈ P(k) be connected. If v ∈ H ∗ (X × Y ) and w = H ∗ (Y × X) have
degrees r and −r (that is, v∗ : H k (X) → H k+r (Y ), or equivalently v ∈ H 2dX +r (X × Y ), then
2dX
X
trX×Y (v ^ w) =
(−1)i tr (w ◦ v)∗ |H i (X)
t
i=0
where the right hand traces are simply traces of linear maps.
15
(4.8)
Proof. We compute each side for v ∈ H 2dX −i (X) ⊗ H j (Y ) and w ∈ H 2dY −j (Y ) ⊗ H i (X). Let {a` }
be a basis of H i (X) and {a0` } the dual basis of H 2dX −i (X), so that trX (a0` ^ am ) = δ`m . Write v, w
in the form
v=
X
a0` ⊗ b`
(4.9)
ck ⊗ ak .
(4.10)
`
w=
X
k
We the compute the left hand side as
!
trX×Y (v ^ t w) = tr
X
a0` ⊗ b` ^
X
`
ak ⊗ ck (−1)deg(a) deg(c)
(4.11)
k


= trX×Y 
X
p∗ (a0` )q ∗ (b` )p∗ (ak )q ∗ (ck )(−1)d(a)d(c) 
(4.12)
`,k


= trX×Y 
X
p∗ (a0` ak )q ∗ (b` ck )
(4.13)
`,k
=
X
trY (b` c` ).
(4.14)
`
Meanwhile, on the right hand side,
!
X
w∗ ◦ v∗ (a` ) = w∗
trX (ak a0k )bk
(4.15)
k
i
= (−1)
X
trY (bk ck )ak
(4.16)
k
and then we sum.
P X i
This has a more familiar form. First, ∆X∗ is the identity on H ∗ (X), so equals 2d
i=0 πX , where
i
∗
i
∗
i
2d
−i
X
πX is the cohomological correspondence H (X) H (X) ,→ H (X). So πX ∈ H
(X) ⊗ H i (X)
is the ith Künneth projector.
Corollary 4.3. Let u ∈ H 2dX (X × X) be of degree zero. Then
2dX
X
i
trX×X (u ^ ∆) =
(−1) tr u∗ |H i (X) .
(4.17)
i=0
Remark. If u is the graph of f : X → X, then u ^ ∆ is the “fixed points of f ” (with suitable
weights).
16
A more refined version is
2dX −i
trX×X (u ^ πX
) = (−1)i tr(u∗ |H i (X) ).
(4.18)
A key example (idea formulated by Weil, and proved by Grothendieck and others): Let k = Fq
and X ∈ P(k). We have a Frobenius morphism F : Xk → Xk essentially given by
[α0 : . . . : αn ] 7→ [α0q : . . . : αnq ]
(4.19)
that is, the k-extension of the absolute q-Frobenius x 7→ xq on coordinate rings of X. Then
X(Fq ) is the set of fixed points of F X(Fq ) and, more generally, X(Fqm ) is the set of fixed points
of F m .
Theorem 4.4. There exists a Weil cohomology on P(Fq ), namely the étale cohomology X 7→
∗ (X , Q ).
H’et
`
k
An immediate consequence is
trX×X (ΓF ^ ∆) =
X
(−1)i tr(F ∗ |H i (X) )
(4.20)
and the left hand trace is |X(Fq )|:
Lemma 4.5. ΓF and ∆X , embedded in X × X intersect properly (every irreducible component of
ΓF ∩ ∆X has codimension 2d, so are closed points).
Then ΓF ^ ∆X can be computed as a sum of local terms, one for each point in ΓF ∩ ∆X .
Claim. The local terms all have multiplicity one.
The claim follows from a check that
T(P,P ) ΓF ∩ T(P,P ) ∆ = (0)
(4.21)
at every fixed point P .
We then obtain
Corollary 4.6 (Weil Conjectures except Riemann Hypothesis). Define the zeta function

Z(X, t) = exp 
X
m≥1
|X(Fqm )|
tm
m

.
(4.22)
Then we have
Z(X, t) =
2d
X
Y
(−1)i+1
det 1 − tF |H i (X)
.
i=0
17
(4.23)
Proof. Apply Lefschetz:

Z(X, t) = exp 
X
m≥1
=
2d
X
Y
2dX
X
!
tr(F m |H i (X) )
i=0

exp (−1)i
i=0
X
m≥1
tm
m



m
t
tr(F m |H i (X) ) 
m
(4.24)
(4.25)
and then the inner sum is equal to
log
1
.
det(1 − tF |H i (X) )
(4.26)
Combining this expression of Z(X, t) with Poincaré duality, we obtain the functional equation
dX χ(X)
1
Z X, d
= ±q 2 tχ(X) Z(X, t)
q Xt
(4.27)
where χ is the Euler characteristic.
5
Intersection of Cycles
Assume X is smooth and quasiprojective. Recall that Zk (X) consists of linear combinations of
dimension k cycles on X. For any proper map f : X → Y , we can push dimension k cycles forward:
(
[k(Z) : k(f (Z))]f (Z) f generically finite
f∗ (Z) =
0
otherwise
(5.1)
On the other hand, we also defined the flat pullback f ∗ : Z k (Y ) → Z k (X) for flat maps
f : X → Y by f ∗ (Z) = [f −1 (Z)]. We would like to generalize this for more functions f , and have
them correspond to cohomological pushforward and pullback, with cup product being intersection
product.
Naively, we’d want an intersection pairing ( . ) : Z k (X) → Z ` (X) → Z k+` (X). We first restrict to
properly intersecting cycles; that is, Z1 , Z2 such that Z1 ∩Z2 has all of its components of codimension
k + `. To define Z1 .Z2 , we want a sum of irreducible components of Z1 ∩ Z2 with some multiplicities.
The correct multiplicity (due to Serre): for each irreducible component W of Z1 ∩ Z2 having
corresponding local ring A, include [W ] in Z1 ∩ Z2 with multiplicity
X
(−1)i lengthA TorA
A/I(Z
),
A/I(Z
)
.
1
2
i
(5.2)
Serre’s formula defines the intersection product for properly intersecting cycles. It remains to
deal with other intersections, such as self-intersections. The classical approach is to “move” one of
18
Z1 and Z2 so that they intersect properly. Specifically, we replace Z1 with a rationally equivalent
cycle.
Rational equivalence generalizes linear equivalence of divisors. We say that cycles Z, Z 0 ∈ Zk (X)
are rationally equivalent if Z − Z 0 is generated by terms of the following form: given a diagram
f
W
η
(5.3)
τ
W
X
with W a closed (integral) subvariety of dimension k + 1, f ∈ k(W )× , and η the normalization
f.
map, take τ∗ (div(f )), with div(f ) considered as a divisor in W
Alternatively, consider a diagram
X × P1
W
(5.4)
P1
with W (k + 1)-dimensional. Cycles rationally equivalent to zero are combinations of [W{0} ] −
[W{∞} ].
Fact. These definitions are equivalent.
The second definition can be generalized to coarser equivalence relations between cycles, such
as algebraic equivalence. H ∗ -homological equivalence is coarser still, and numerical equivalence is
possibly even coarser (conjecturally equal to homological).
The Chow group (implicitly with Q coefficients) is
CH k (X) = Z k (X)Q / ∼rat .
(5.5)
In order to obtain a well-defined intersection theory CH k (X) × CH ` (X) → CH k+` (X), we need
Lemma 5.1 (Chow’s Moving Lemma). (1) For X smooth and quasiprojective and cycles Z1 , Z2
on X, there exists Z 0 rationally equivalent to Z1 such that Z 0 and Z2 intersect properly.
(2) If Z 0 , Z 00 are both rationally equivalent to Z1 such that Z 0 and Z 00 each intersect Z2 properly,
then Z 0 .Z2 and Z 00 .Z2 are rationally equivalent.
Corollary 5.2. CH ∗ (X) is a commutative, graded ring.
We get formalism of f∗ for proper maps and f ∗ for maps f : X → Y with Y proper (more
general cases are also possible). In this situation,
f ∗ (Z) = pX∗ (Γf .[p∗Y (Z)])
where p∗Y is a flat pullback, so has already been defined.
19
(5.6)
6
Adequate Equivalence Relations
We will define equivalence relations which will be useful in defining a category of pure motives
modulo equivalence.
An adequate equivalence relation is, for every X ∈ P(k), an equivalence relation ∼ on Z ∗ (X)
such that:
• ∼ respects the linear structure.
• Z ∗ (X)/ ∼ is a ring under intersection product. Specifically, there is an analogue of Chow’s
Moving Lemma for ∼ inducing a well-defined ring structure.
• For every proper map f : X → Y , if α ∼ 0, then f∗ α ∼ 0. Hence we obtain a map f∗ :
Z ∗ (X)/ ∼→ Z ∗ (Y )/ ∼.
• Similarly, the pullback induced by
f ∗ (β) = p∗ (Γf .q ∗ (β))
(6.1)
is well-defined modulo ∼.
• f∗ and f ∗ are related by the projection formula
f∗ (α.f ∗ β) = f∗ (α).β
(6.2)
Some examples are:
1. Rational equivalence is adequate.
2. For every Weil cohomology H ∗ , write α ∼H ∗ β if γX (α) = γX (β) in H ∗ (X). This is an adequate
equivalence relation. A priori, ∼H ∗ depends on the choice of H ∗ , but if two cohomologies are
related by a comparison isomorphism, then the equivalence relations are identical.
3. Define numerical equivalence on Z ∗ (X) as follows: α ∈ Z k (X) has α ∼num 0 if for every
β ∈ Z d−k (X), deg(α.β) = 0. Here the degree map Z d (X) → Z is given by
X
ni Pi 7→
X
ni [k(Pi ) : k].
This map factors through the Chow group. We might interpret this relation as
Lemma 6.1.
(6.3)
R
X
α.β = 0.
(1) Rational equivalence is the finest adequate equivalence relation.
(2) Numerical equivalence is the coarsest adequate equivalence relation.
To show (2), consider the degree map as π∗ : Z d (X) → Z 0 (Spec k) arising from the structure
map π : X → Spec k..
20
Proof of (1). Let ∼ be an adequate equivalence relation and suppose α ∼rat 0. Then there is a
diagram
P1 × X
Wi
(6.4)
p
P1
such that
α=
=
X
X
[Wi (0)] − [Wi (∞)]
q∗ (p∗ ([0] − [∞]).Wi )
(6.5)
(6.6)
i
Because ∼ is adequate, to show α ∼ 0 it’s enough to show [0] ∼ [∞].
For simplicity, let k P
= k. Let x, y ∈ P1 (k). As ∼ is adequate, there exists z ∼ x intersecting
properly with x, so z =
zi with zi 6= x. Write down a map f : P1 → P1 such that f∗ (x) = y and
f∗ (zi ) = zi . Such a map is
f (T ) = T + (y − x)
Y T − zi
.
x − zi
(6.7)
Given such a map f ,
x ∼ z ∼ f∗ (z) ∼ f∗ (x) ∼ y.
(6.8)
Let ∼ be an adequate equivalence relation on Z ∗ (X) (we will implicitly use Q coefficients).
Let E be a field of characteristic zero, to be used as a possibly larger field of coefficients. Let
A∗ (X) = Z ∗ (X)E / ∼ be the ring of cycles modulo ∼. The composition law for cohomological
correspondences works as well for the ∼ correspondences. That is, there are maps
AdY +s (Y × Z) × AdX +r (X × Y )
(β, α)
Adx+r+s (X × Y )
h
i
pXZ∗ p∗XY (α).p∗Y Z (β)
(6.9)
In particular, AdX (X × X) is a ring, which will end up being the endomorphisms of X as a
motive modulo ∼.
We now define the category M∼
k of pure motives over k modulo ∼, with E implicit. Begin with
P(k), and consider the category Corr∼ (k) with objects the varieties X (usually denoted h(X)) and
the morphisms given by
Hom(h(X), h(Y )) = AdX (X × Y ).
21
(6.10)
This is an E-linear category, with h(X) ⊕ h(Y ) = h(X q Y ).
There is a functor
P(k)op
Corr∼ (k)
X
h(X)
f
tΓ
f
X−
→Y
(6.11)
where, if Γf ∈ Z dY (X × Y ), its transpose lies in Z dY (Y × X). We would like to enlarge Corr∼
to include images of projectors. There’s a universal way of doing this, called the pseudo-abelian
envelope. We also want duals to exist in our theory, which amounts to adding Tate twists.
We combine these two steps into one and define M∼
k to have objects X × P(k) along with an
idempotent element p ∈ AdX (X × X) and an integer n ∈ Z (considered as a Tate twist). [The object
(X, p, n) should be thought of as pH ∗ (X)(n).] The set of maps is
Hom((X, p, n), (Y, q, m)) = qAdX +m−n (X × Y )p.
(6.12)
M∼
k has the following properties:
• It is pseudo-abelian.
• It is E-linear. One needs to think about how (X, p, n) ⊕ (Y, q, m) is defined. Here is the idea:
there is a canonical isomorphism from (Spec k, 1, −1) to a summand of h(P1 ) (which should
be thought of as H 2 (P1 )), denoted 1(−1). Then (assuming n ≥ m) we define
(X, p, n) ⊕ (Y, q, m) =
!
X q Y × (P1 )n−m , p ⊕ q 0 , n
(6.13)
where we identify (Y, q, m − n) with (Y × (P1 )n−m , q 0 , 0).
• (To be shown a bit later) It is a ⊗-category with h(X) ⊗ h(Y ) = h(X × Y ). For morphisms,
take external product and swap factors suitably.
Grothendieck conjectured:
Conjecture 6.2 (Standard Conjecture D(X)). For X ∈ P(k) and H ∗ any Weil cohomology, ∼H ∗
coincides with ∼num .
This is the most important standard conjecture, and in characteristic zero implies the rest. He
∗
also conjectured that Mnum
is abelian (hence D would imply MH
k
k is abelian). In the early 90’s, it
was proven that:
Theorem 6.3 (Jannsen). Let ∼ be an adequate equivalence relation. Then the following are equivalent:
(1) M∼
k is semisimple abelian.
22
(2) For every X, Adx (X × X) is a finite dimensional semisimple E-algebra.
(3) ∼ coincides with ∼num .
Proof. (1) =⇒ (3): Recall ∼num is the coarsest adequate equivalence relation, so we always have
•
a surjection A•∼ (X)
R Anum (X). We need to show injectivity. That is, if f 6= 0, we need to find a
cycle g such that X f.g 6= 0.
Let f ∈ Aj (X); this is a morphism f : 1 → (X, 1, j). If f 6= 0, then f must be mono (because
End(1) = E is a field). Semisimplicity impliesR there exists g : (X, 1, j) → 1 as an element of
AdX −j (X) such that g ◦ f is the identity. Then X g.f = 1 6= 0.
(3) =⇒ (2): We consider Anum (X) with Q-coefficients. Choose any Weil cohomology H ∗ (for
example, we could always choose `-adic cohomology with ` 6= 0) with E-coefficients, and let A•H (X)
be the cycles modulo H ∗ -homological equivalence. We have a surjection A•H (X) A•num (X)E , and
X (X × X)
therefore a surjection S : AdHX (X × X) Adnum
num . The first ring is finite-dimensional, so
the second ring also is.
Let JH and J be the Jacobson radicals; we need to show that J = 0. Because we have a
X /S(J ). The first
surjection, we get a map JH → J, and therefore a surjection AdHX /JH Adnum
H
ring is finite dimensional and semisimple, so the second ring is. We get S(JH ) = J.
Let f ∈ J; lift it to fH ∈ JH . For every g ∈ A∗H (X × X), fH .g ∈ JH , so is in particular nilpotent.
Now apply the trace formula:
trX×X (g.fH ) =
X
(−1)i tr(fH ◦ g)|H i (X)
= 0.
(6.14)
(6.15)
Since g was arbitrary, f = S(fH ) is numerically trivial. Hence J = 0.
(2) =⇒ (1): This is purely formal. Since any object of M∼
k is, after some Tate twist, a direct
summand of some (X, 1, 0), we find that for every A ∈ M∼
,
End
Mk (M ) is finite dimensional and
k
semisimple.
Explicitly, if (X, 1, 0) = M ⊕ N and if J ⊆ End(M ) were a 2-sided nilpotent ideal, then
J
Hom(M, N ) ◦ J
J ◦ Hom(N, M )
Hom(M, N ) ◦ J ◦ Hom(N, M )
(6.16)
would also be a nonzero 2-sided ideal, a contradiction. So it remains to show:
Lemma 6.4. If C is any E-linear pseudo-abelian category such that for every M , End(M ) is finitedimensional semisimple, then C is semisimple abelian.
Proof. Wedderburn’s theorem implies all End(M ) are direct sums of matrix algebras over finite
dimensional division algebras over E, and M is indecomposable if and only if End(M ) is a division
algebra.
Take N, M indecomposable. Then we claim either N ∼
= M or Hom(N, M ) = 0. This implies the
result because then C is equivalent to a direct sum of categories of End(M )-vector spaces indexed
by the indecomposable M .
23
Suppose we had g ∈ Hom(M, N ) nonzero. Then consider the composition
Hom(M, N ) × Hom(N, M )
End(M )
(g, f )
g◦f
(6.17)
We
claim that either this composition or the reverse (f, g) 7→ f ◦ g ∈ End(N ) is nonzero. If not,
0
0
then Hom(M,N
) 0 is a nonzero 2-sided nilpotent ideal of End(M ⊕ N ), a contradiction.
So there is a nonzero composition, implying there exist g, f such that g ◦ f is nonzero in the
division algebra End(M ). Now h = f ◦ (g ◦ f )−1 is a right inverse of g. In particular, h ◦ g is an
idempotent in End(N ). By indecomposability of N , h◦g must be an isomorphism. This would imply
g is an isomorphism.
is abelian.
As a consequence, Mnum
k
7
Tannakian Theory
is
For a fixed Weil cohomology, we would like to obtain a “motivic Galois formalism” where Mhom
k
equivalent to the category of representations of some large group.
Let (C, ⊗) be an additive tensor category. (The ⊗ operation is associative, commutative, and
has a unit, up to functorial isomorphism. This is also referred to as a symmetric monoidal category.)
Check that for a unit object 1, E = End(1) is a commutative ring, and C is E-linear.
X∨
We say that C is rigid if for every object X, there exists an object X ∨ and morphisms evX :
⊗ X → 1 and coevX : 1 → X ⊗ X ∨ such that the composites
X
X∨
coevX ⊗idX
X ⊗ X∨ ⊗ X
idX ⊗evX
idX ∨ ⊗coevX
X∨ ⊗ X ⊗ X∨
evX ⊗idX ∨
X
(7.1)
X∨
are both the identities.
Fact. The rigidity condition gives internal homs. That is,
T 7→ Hom(T ⊗ X, Y )
(7.2)
is representable by an object Hom(X, Y ) (namely Y ⊗ X ∨ , because then given T → Y ⊗ X ∨ ,
we get T ⊗ X → Y ⊗ X ∨ ⊗ X → Y ).
Let E be a field. A neutral Tannakian category C over E is a rigid abelian ⊗-category with
End(1) = E and for which there exists a fiber functor ω : C → VectE (that is, a faithful, exact,
E-linear ⊗-functor).
Theorem 7.1. Let (C, ⊗) be a neutral Tannakian category over E, and let ω : C → VectE be a fiber
functor. Then the functor on E-algebras given by Aut⊗ (ω) is represented by an affine group scheme
over E, and C → RepE (Aut⊗ (ω)) is an equivalence.
24
The functor Aut⊗ : E − alg → Set maps R to collections of (gX )X∈C such that:
1. gX is an R-linear automorphism ω(X) ⊗ R.
2. Given f : X → Y , the diagram
gX
ω(X) ⊗ R
ω(X) ⊗ R
(7.3)
gY
ω(Y ) ⊗ R
ω(Y ) ⊗ R
commutes. Also we have a commutative diagram
(ω(X) ⊗ R) ⊗R (ω(Y ) ⊗ R)
ω(X ⊗ Y ) ⊗ R
gX⊗Y
gX ⊗gY
···
(7.4)
···
3. We require that g1 is the identity on ω(1) ⊗ R.
, with its given commutativity constraint, is not
However, even if we assume ∼hom =∼num , Mhom
k
Tannakian.
Any rigid ⊗-category has an intrinsic notion of rank. For X ∈ C, the composite
cX,X ∨
coev
ev
1 −−−−X
→ X ⊗ X ∨ −−−−→ X ∨ ⊗ X −−X
→1
(7.5)
in End(1) is called the rank of X. For example, in VectE , let vi and vi? be a basis and dual basis,
and then we have
1 7→
In GrE , the rank is
P
X
vi ⊗ vi? 7→
X
vi? ⊗ vi 7→ dim V
(7.6)
(−1)i dim V i where V i is the ith graded piece.
Now any ⊗-functor preserves rank. Using the functor H ∗ : Mhom
→ GrE , we see that Mhom
k
k
has objects of negative rank, so cannot have a fiber functor.
Lemma 7.2. Let k be of characteristic zero and H ∗ a Weil cohomology. Assume that ∼hom =∼num
i ∈ H ∗ (X × X) are algebraic classes. Then Mhom is a neutral Tannakian category
and that every πX
k
over E.
Proof. By Jannsen’s theorem, Mhom
= Mnum
is abelian. We modify the commutativity constraint
k
k
using the Künneth conjecture (C(X) for every X). Künneth implies Mk is Z-graded via the projectors π i . That is, for every M ∈ Mk , we get a “weight decomposition”
M=
M
i∈Z
25
π i M.
(7.7)
For every M, N , define the modified commutativity constraint c0M,N =
i
j
j
i
ci,j
M,N : π M ⊗ π N → π N ⊗ π M.
L
ij i,j
i,j (−1) cM,N
for
(7.8)
Then H ∗ : Mk → VectE is a fiber functor.
8
Standard Conjectures
The Künneth Conjecture C(X)
8.1
i ∈ H ∗ (X×X)
Conjecture 8.1 (Künneth; Standard Conjecture C(X)). For every i, the projector πX
is algebraic.
Here are some examples:
1. For any k, if X is an abelian variety, then C(X) is true.
2. Let k be a finite field. Then for every X, C(X) is true (for every Weil cohomology satisfying
weak Lefschetz).
This is a theorem of Katz-Messing. The idea is that for X/Fq , we have a Frobenius map
F : XFq → XFq . Deligne showed F H`i (XF2 ) is pure of weight i. meaning the eigenvalues of
i
F all have absolute value q 2 under every isomorphism Q` ∼
= C. In particular,
P i (t) = det(1 − F t|H i (X) )
`
(8.1)
lie in Q[t] and are coprime for different i. Katz-Messing shows that for any H ∗ (with weak
Lefschetz), P i (t) = det(1 − F t|H i (X) ) as well. So choose a polynomial Πi (t) ∈ Q[t] such that
for j 6= i, P j (t) divides Πi (t), and Πi (t) ≡ 1 (mod P i (t)). Then by Cayley-Hamilton, the
algebraic class Πi (F ) is the ith projector.
In general, fix H ∗ . Let A• (X ×X) ⊆ H 2• (X ×X)(•) be the image of the cycle map on Z • (X ×X)
(with Q-coefficients).
Lemma 8.2. (1) A• (X × X) is a Q-subalgebra of H 2• (X × X) with the algebra structure on
H ∗ (X × X) given by that of End(H ∗ (X)).
(2) For every algebraic class a ∈ H 2r (X), the map
^ a : H ∗ (X) → H •+2r (X)
is algebraic (it lies in AdX +r (X × X)).
Proof.
(1) We’ve seen this already.
26
(8.2)
(2) This follows from ^ a being the map on cohomology associated to ∆∗ (a). Indeed,
∆∗ (a) (b) = q∗ (p∗ b.∆∗ (a))
(8.3)
= q∗ ((b ⊗ [X]).∆∗ (a))
(8.4)
∗
= q∗ ∆∗ (∆ (b ⊗ [X]).a)
(8.5)
= 1∗ (b.a)
(8.6)
= b ^ a.
(8.7)
Here is another consequence of the Künneth conjecture: for every u ∈ Z d (X × X),
2d−i
tr(u∗ |H i (X) ) = (−1)i deg(u.πX
)
(8.8)
will be rational. (A priori the trace only lies in E.) Hence the minimal polynomial of u∗ H i (X)
has Q coefficients, and (?) if u is an isomorphism of H i (X), then u−1 is also algebraic (because then
u−1 ∈ Q[u]).
8.2
The Lefschetz Conjecture
We say that a Weil cohomology satisfies the hard Lefschetz theorem if, for every X ∈ P(k), if η is
an ample line bundle, the operator
L = Lη = ^ c1 (η) : H ∗ (X) → H •+2 (X)(1)
(8.9)
is such that
∼
Ld−i : H i (X) −
→ H 2d−i (X)(d − i)
(8.10)
for every i ≤ d.
∗ the Betti cohomology, this is part of Hodge theory. For any k and
For k = C and H ∗ = HB
H ∗ = H`∗ , this is due to Deligne (in Weil II).
Under the hard Lefschetz theorem, there is a primitive decomposition of H ∗ (X): for every i ≤ d,
we write
Primiη (X) = ker(Lηd−i+1 ).
(8.11)
Then we can decompose
H ∗ (X) =
d−j
d M
M
Ljη Primiη (X).
(8.12)
i=0 j=0
L H ∗ (X) is a nilpotent operator. The Jacobson-Morosov theorem implies the action of L
extends to a representation of sl2 H ∗ (X) where L corresponds to e = ( 00 10 ). The algebra sl2 is
27
0 , and f = ( 0 0 ). We then decompose H ∗ (X) into lowest weight spaces
generated by e, h = 10 −1
10
and then separate into isotypic components.
Theorem 8.3 (Jacobson-Morosov). Let E be a field (maybe we require perfect; certainly characteristic 0 suffices). Let g be a semisimple Lie algebra over E, x ∈ g be a nonzero nilpotent element.
Then:
(1) There exists an sl2 -triple in g extending x. meaning (x, y, h) satisfying [h, x] = 2x, [h, y] =
−2f , and [x, y] = h.
(2) Given x, for any semisimple element h such that [h, x] = 2x, there exists a unique y making
(x, y, h) an sl2 -triple.
As an application of (2), we can start with L and
2d
X
i
Π=
(i − d)πX
.
(8.13)
i=0
Then Π is semisimple and
Π ◦ L − L ◦ Π)(v) = (i + 2 − d)Lv − (i − d)Lv
(8.14)
= 2Lv
(8.15)
[Π, L] = 2L.
(8.16)
So there exists a unique operator c Λ making (Π, L, c Λ) an sl2 -triple. Now
Primiη (X) = ker(c Λ|H i (X) ).
(8.17)
Proof of Jacobson-Morosov. First we show uniqueness in (2). Given triples (x, y, h) and (x, y 0 , h),
[x, y − y 0 ] = 0 so y − y 0 ∈ Cent(x). But also y − y 0 is contained in the (−2)-eigenspace for ad(h).
This forces y − y 0 = 0 since highest weight spaces in sl2 -theory have nonnegative h-eigenvalues.
Next, we consider existence. Suppose for now that hL∈ [g, x]. We can then write h = [x,
P y]
gλi into h-eigenspaces. Let y =
yi
for some y. h is semisimple, so we can decompose g =
accordingly. Then
g0 3 h =
X
[x, yi ]
| {z }
(8.18)
∈gλi +2
so h = [x, y−2 ] and (x, y−2 , h) is an sl2 -triple.
Now it remains to show that for any semisimple h such that [h, x] = 2x, we have h ∈ [g, x].
We induct on dim g. We may assume g 6= sl2 , so dim g > 3, and then assume that x lies in no
proper semisimple subalgebra of g. First, [g, x] = Cent(x)⊥ where the orthogonal complement is
with respect to the Killing form κ. Indeed, if c ∈ Cent(x),
κ([g, x], c) = κ(g, [x, c]) = 0
28
(8.19)
so we get inclusion. Equality then holds by nondegeneracy of κ. So if h ∈
/ [g, x], then h ∈
/
⊥
Cent(x) , so κ(h, Cent(x)) 6= 0. We decompose Cent(x) into h-eigenspaces Cent(x)λi . Now there
exists z ∈ Cent(x)0 such that κ(h, z) 6= 0, for
0 = κ([h, h], zλi )
(8.20)
= κ(h, [h, zλi ])
(8.21)
= λκ(h, zi )
(8.22)
λi = 0.
(8.23)
Now z cannot be nilpotent, by considering 0 6= κ(h, z) = tr(ad(h) ◦ ad(z)). This would be zero
if z were nilpotent and commuting with h. So now z s , the semisimple part of z, is nonzero. Then
Cent(z s ) is a reductive, proper Lie subalgebra. Now [Cent(z s ), Cent(z s )] is a proper semisimple
subalgebra of g, containing x. This is a contradiction. Thus h ∈ [g, x].
Finally, to prove (1), we need to show that given a nilpotent x, there exists a semisimple h such
that [h, x] = 2x. Recall Cent(x)⊥ = [g, x] But for x nilpotent, κ(x, Cent(x)) = 0. In fact, if z and x
commute, then ad(x) and ad(z) commute, so their composition is nilpotent, hence has trace 0.
Thus there exists some h ∈ g such that [h, x] = 2x. We can take h to be semisimple since if
h = hs + hn , then [hn , x] = 0, so [hs , x] = 2x.
We return to our application. Any x ∈ H j (X) may be written as
X
x=
Lk xj−2k
(8.24)
k≥max(0,−d)
for xj−2k ∈ Primj−2k (X). Then
c
Λ(x) =
X
k(d − j + k + 1)Lk−1 xj−2k .
(8.25)
k
We can extract a convenient “Hodge-star” operator: given our representation sl2 H ∗ (X), we
0 1 . If
obtain a representation SL2 H ∗ (X). Inside SL2 , we have the Weyl group element w = −1
0
v ∈ H ∗ (X) lies in the weight i eigenspace for Π, then wv lies in the (−i)-eigenspace. Also w2 = −1.
We renormalize w on each degree to obtain
?H = (−1)
j(j+1)
2
w : H j (X) → H 2d−j (X).
(8.26)
Now ?2H = 1. A variant is the Lefschetz involution
?L (x) =
X
Ld−j+k xj−2k .
(8.27)
These two involutions are related by
?H (x) =
X
(−1)
(j−2k)(j−2k+1)
2
k
29
k!
Ld−j+k xj−2k .
(d − j + k)!
(8.28)
There is also a cohomological correspondence Λ which is inverse to L on the image of L. Formally
Λ = ?L L?L . We also introduce the primitive projectors pj for 0 ≤ j ≤ 2d. For 0 ≤ j < d, pj is zero
outside degree j, and pj (x) = xj for x ∈ H j (X). For d < j ≤ 2d, pj = p2d−j Λj−d .
Lemma 8.4. The operators Λ, c Λ, ?L , ?H , π 0 , . . . , π 2d , and p0 , . . . , pd−1 are all given by universal
noncommutative polynomials over Z in L and pd , . . . , p2d .
Proof. This follows from working in the primitive decomposition x =
each xj−2k as
P
Lk xj−2k and extracting
xj−2k = p2d−j+2k Ld−j+k x.
(8.29)
Corollary 8.5. The following Q-subalgebras of End H ∗ (X) are equal:
Q[L, Λ] = Q[L, c Λ] = Q[L, ?L ] = Q[L, ?H ] = Q[L, pd , . . . , p2d ]
(8.30)
and they all contain p0 , . . . , pd−1 and π 0 , . . . , π 2d .
This will follow from the above lemma and:
Lemma 8.6. Q[L, Λ] contains pd , . . . , p2d .
We now state various forms of the Lefschetz Standard Conjecture:
Conjecture 8.7 (Weak form, A(X, L)). For 2p ≤ d, Ld−2p : Ap (X)toAd−p (X) is an isomorphism
(specifically, surjective).
Conjecture 8.8 (Strong form, B(X)). The operator Λ : H ∗ (X) → H •−2 (X)(−1) is algebraic. That
is, Λ is the cohomology class of some cycle in Adx −1 (X × X).
Proposition 8.9. The following are equivalent:
(1) A(X, L).
(2) A• (X) is stable under pd , . . . , p2d .
(3) A• (X) is stable under ?H (or ?L ).
(4) A• (X) is stable under Λ or c Λ.
In particular, B(X) implies A(X, L).
Proposition 8.10. The following are equivalent:
(1) B(X).
(2) pd , . . . , p2d are algebraic.
(3) ?H (or ?L ) is algebraic.
30
(4) Λ or c Λ is algebraic.
(5) For every i ≤ d, the inverse of Ld−i is algebraic.
[(5) =⇒ (1) uses something not yet written down.]
Corollary 8.11. B(X) implies C(X).
i is contained in Q[L, Λ].
Proof. Every πX
B(X) also implies a stronger result than the consequence (?) of C(X): if u ∈ AdX +
that
i
j
^ u : H (X) → H (Y )
j−i
2
j−i
2
(X) such
(8.31)
is an isomorphism, then u−1 is algebraic. For a map H j (Y ) → H i (X) is v = ?L,X ◦ t u ◦ ?L,Y .
Under B(X) and B(Y ), v is an algebraic isomorphism. Now v ◦ u is an algebraic isomorphism
H i (X) → H i (X), so we may apply (?).
Corollary 8.12. B(X) is independent of the choice of ample line bundle η such that L = ^ c1 (η).
Proof. If we had another L0 = Lη0 , then hard Lefschetz implies (L0 ) : H i (X) → H 2d−i (X) is an
algebraic isomorphism. Then B(X) (using L!) implies its inverse is algebraic.
8.3
The Hodge Standard Conjecture
(This is not the same thing as the Hodge Conjecture.)
k (X) carries a Q-pure Hodge structure
Take our field to be C and X ∈ P(C). For every k ≥ 0, HB
of weight k. More fundamental in algebraic geometry are the polarizable Q-Hodge structures.
On X, let L be an ample line bundle. Then c1 (L) ∈ H 2 (X, Q)(1). Let ω =
which may be considered as a Kähler form in H 2 (X, R). Now we have
H k (X, R) × H k (X, R)
Qω
α, β
R
c1 (L)
2πi
∈ H 2 (X, Q),
R
(8.32)
d−k
Xα^β ^ω
0
0
(we could also work with Q coefficients). Extending C-linearly, we see that Qω (H p,q , H p ,q ) = 0
unless p + p0 = q + q 0 , so that (p, q) = (q 0 , p0 ). In other words, the sesquilinear pairing
H k (X, C) × H k (X, C)
C
ik Q
α, β
0
Hω
0
(8.33)
ω (α, β)
satisfies Hω (H p,q , H p ,q ) = 0 unless (p, q) = (p0 , q 0 ). Also, different pieces of the primitive decomposition are orthogonal: if α = Lrω α0 and β = Lsω β0 for α0 , β0 primitive and r 6= s, then
Hω (α, β) = 0. Otherwise, we may assume r < s. Then
31
Z
Qω (α, β) =
Lωd−k+r+s α0 ^ β0
(8.34)
X
and r + s ≥ 2r + 1 implies Lωd−k+r+s α0 = 0.
We want to study the positivity properties of Hω , reducing to particular pieces of the bigrading
and the primitive decomposition. We write Primp,q = Prim ∩ H p,q .
Theorem 8.13 (“Hodge index theorem”). On Lrω Primp,q (X) ⊆ H k (X, C), Hω is definite with sign
(−1)
k(k−1)
2
ip−q−k .
For example, if X is a curve, Hω has sign −1 on H 0,1 and 1 on H 1,0 . For we have
Z
Hω (α, α) = i
α ∧ α.
(8.35)
|f |2 dx ∧ dy > 0.
(8.36)
X
Locally, α = f (z) dz for z = x + iy, then this is
Z
i
Z
f (z) dz ∧ f (z) dz = 2
If X is a surface, then Hω has sign 1 on Prim1,1 and −1 on H 2,0 ⊕ H 0,2 . Thus Qω has sign −1
1,1
on PrimR
and 1 on (H 2,0 ⊕ H 0,2 )R . On the other hand, Qω is positive definite on Rω, so using the
primitive piece was necessary.
If X is a K3 surface, then the signature on Qω is (2, 19) on Prim2 (X) and (3, 19) on H 2 (X).
A weight k Q-Hodge structure is a (V, h) for V a Q-vector space and h a representation on VR
of S = ResC/R Gm . Explicitly, S(R) = C× , and z acts on H p,q as z p z q .
A Q-Hodge structure is polarizable if there exists a morphism of Hodge structures Q : V ⊗ V →
Q(−k) such that (2πi)k Q(v, h(i)w) : VR × VR → R is positive definite.
Corollary 8.14 (of Hodge index theorem). For every X/C smooth projective, Primk (X, Q) is a
polarizable Q-Hodge structure. A polarization is given by (−1)
k(k+1)
2
times
Q
Primk (X) × Primk (X)
Q(−k)
1
(2πi)d
α, β
(8.37)
d−k α ^ β
XL
R
where L arises from the ample line bundle η.
Proof. We need to show that for ω =
on Primk (X, R) = VR .
c1 (η)
2πi
∈ H 2 (X, Q), the pairing Qω (, h(i)) is positive definite
Let v ∈ VR be of the form v p,q + v q,p with v p,q = v q,p . Then
32
Qω (v, h(i)v) = i−k Hω (v, h(i)v)
(8.38)
= i−k Hω (v p,q + v q,p , ip−q v p,q + iq−p v q,p )
=i
−k
p,q
q,p
q−p q,p
(8.39)
p−q p,q
Hω (v + v , i v + i v )
= i−k iq−p Hω (v p,q , v p,q ) + ip−q Hω (v q,p , v q,p ) .
(8.40)
(8.41)
Now apply the Hodge index theorem to Hω for both terms to get the desired sign.
We can also polarize H ∗ (X, Q) by variants of Q with some sign changes.
We want a (weaker) version of this result that works over any field and any Weil cohomology.
Conjecture 8.15 (I(X)). If 2p ≤ d = dim X, the pairing
Ap (X) ∩ Prim2p (X)(p) × Ap (X) ∩ Prim2p (X)(p)
Q
(8.42)
(−1)p trX (Ld−2p x.y)
x, y
is positive definite.
∗.
Corollary 8.16 (of Hodge index theorem). I(X) is true for k = C and H ∗ = HB
If k is a finite field, this is related to the Riemann Hypothesis for P(k): first, we reformulate
I(X) as saying that the pairing
A• (X) × A• (X)
Q
x, y
trX (x. ?H y)
(8.43)
is positive definite. In this case, then on the algebra of algebraic correspondences AdX (X × X),
we get a positive involution given by
hαv, ?H wi = hv, ?H α0 wi
(8.44)
for hx, yi = trX (x.y). This holds for α0 = ?H t α?H , which is algebraic assuming B(X).
Now given X/k of dimension d, we have the Frobenius F : Xk → Xk , and want to know that
the eigenvalues of F H i (X) are pure of weight i. We renormalize by defining
Φ=
2d
X
i
√
i
q − 2 πX
F ∗ H ∗ (X)[ q].
(8.45)
i=0
√
Under B(X), Φ ∈ AdX (X × X) ⊗Q Q[ q]. We want to show that all of the eigenvalues of Φ
have absolute value 1 in all complex embeddings. We will obtain this by realizing Φ as a unitary
operator on some inner product space.
A Φ commutes with ?H , we have Φ0 = t Φ.
33
Claim. t Φ = Φ−1 , so that h , ?H i is invariant under Φ.
To show this, take g = Φ in:
Lemma 8.17. If g ∈ End(H ∗ (X)) is such that:
(1) deg g = 0.
(2) g(a.b) = g(a).g(b).
(3) g(η) = η.
then g is invertible and g −1 = t g.
Proof. (2) and (3) imply that g|H 2d is the identity. This implies g is an automorphism: given a
nonzero a ∈ H i , find b ∈ H 2d−i such that trX (a.b) 6= 0, so that g(a).g(b) = a.b which has a nonzero
trace; hence g(a) 6= 0.
Now, for a ∈ H i and b ∈ H 2d−i ,
hg −1 (a).bi = hg(g −1 a.b)i
= ha.g(b)i
(8.46)
(8.47)
so that g −1 = t g.
√
√
Now Q[ q][Φ] ⊆ AdX (X × X)[ q] Φ, the action being α 7→ α ◦ Φ. It is unitary with respect to
√
the inner product (α, β) = tr(α.β 0 ). So all eigenvalues of Φ acting on Q[ q][Φ] have absolute value
1, implying the same result holds for the roots of the characteristic polynomial of Φ.
8.4
Dependencies
(1) B(X) and I(X) together imply D(X). In particular, in characteristic 0, B(X) implies every
standard conjecture.
(2) D(X × X) implies B(X).
deg
To show (1), if the pairing Ap (X) × Ad−p (X) −−→ Q were nondegenerate, then D(X) would
follow. We know from I(X) that h, ?H i is nondegenerate, and then B(X) implies the first pairing
is nondegenerate.
(2) follows from an argument of Smirnov.
9
Motivated Cycles
Our goal is to construct a modified category Mk of “pure motives modulo homological equivalence”
such that:
34
1. Mk ' Mhom
provided the standard conjectures are true.
k
2. Mk has all of the categorical properties we want. Say k has characteristic 0; then we have a
Q-linear, semisimple, neutral Tannakian category. (This gives an unconditional motivic Galois
formalism.)
3. Mk allows us to:
(a) Prove some unconditional results.
(b) Formulate some interesting problems which are hopefully more tractable than the standard conjectures.
The basic strategy is to modify our characterization of “nice” cycles:
algebraic ⊆ motivated (André) ⊆ absolute Hodge (Deligne) ⊆ Hodge
We can redefine the algebra of correspondences by using one of these larger classes of cycles.
An absolute Hodge cycle in X/k for k algebraically closed and of finite transcendence degree
(and X smooth and projective) is a class t in
2p
HdR
(X)(p) × HA2p (X)(p)
(9.1)
n
b ⊗Z Q,
(X, Z)
HAn (X) = Hét
(9.2)
where we write
∗ : H 2p (X)(p) → H 2p (σX)(p) and
such that for every embedding σ : k ,→ C, which give σdR
dR
dR
∗ , we get σ ∗ t ∈ H (σX) × H (σX). By Betti-de Rham and Betti-de Rham
a similar pullback σét
A
dR
2p
(σX)(p). t is absolute Hodge if σ ∗ t comes from a
comparison isomorphisms, we may embed HB
2p
rational class of type (0, 0) in HB (σX)(p).
Theorem 9.1 (Deligne). For k of characteristic 0, any Hodge cycle on an abelian variety is absolutely Hodge.
This should be thought of as a weakening of the Hodge conjecture for abelian varieties. This has
classical applications:
1. Algebraicity of (products of) special values of the
with refinements giving the Galois
Q Γ function,
ai
?
action. Roughly, these have the form (2πi)
i∈? Γ( d ) ∈ Q. Roughly, this originates from
considering the periods (coefficients appearing in a Betti-de Rham comparison isomorphism)
of the Fermat hypersurface
d
X0d + · · · + Xn+1
= 0.
(9.3)
∗ (X/k),
For an
Z ,→ XC , then for a differential form ω such that [ω] ∈ HdR
R algebraic cycle
?
then Z ω ∈ (2πi) k. The products of Γ’s arise as periods on the Fermat hypersurface. The
same principle applies when Z corresponds to an absolute Hodge cycle. We can use Hodge
cycles instead, by the theorem for abelian varieties (the Tannakian subcategory generated by
abelian varieties contains the cohomology of the Fermat hypersurfaces).
35
2. A very weak half of a case of the Grothendieck period conjecture: suppose X/k is smooth and
projective over a number field. The Betti-de Rham period matrix gives a well-defined double
coset of GLN (k)\GLN (C)/GLN (Q). Let k(ΩX ) be the field generated by the coefficients of
the period matrix. It’s conjectured that
tr degk k(ΩX ) = dim(motivic Galois group of X).
(9.4)
Deligne’s theorem implies that for the motive H 1 (A), with A an abelian variety over k,
tr degk (k(ΩH 1 (A) )) ≤ dim(GH
1 (A)
)
= dim(M T (H 1 (A)))
(9.5)
(9.6)
The Mumford-Tate group is the Hodge theory analogue of the motivic Galois group. Without
Deligne’s theorem, the second equality would have only been ≥.
Compare this with the motives Q and H 2 (P1 ). For Q, the Mumford-Tate group is trivial,
k(Ω) = Q. But for H 2 (P1 ), the Mumford-Tate group is Gm (of dimension 1) and k(Ω) = Q(2πi)
(of transcendence degree 1).
3. (André) Let F be finitely generated over Q and X, X 0 two (polarized) K3 surfaces over F .
Recall that X/C is a K3 surface if:
• H 1 (X, OX ) = 0.
• KX (which is Ω2X ) is isomorphic to OX .
• As a consequence, h2,0 (X) = 1 and h1,1 (X) = 20.
Then any isomorphism of Galois modules Prim2 (X, Q` ) ∼
= Prim2 (X 0 , Q` ) arises from a Q` linear combination of motivated cycles. Also, we have that the Mumford-Tate conjecture is
true for Prim2 (X). The Mumford-Tate conjecture is that
0
M T (Prim2 (XC )) ⊗ Q` ∼
= Zariski closure of the image of ΓF Prim2 (XF , Q` ) .
(9.7)
Meanwhile, the Mumford-Tate group of a K3 surface is quite restricted. Prim2 (X) contains
the Hodge cycles and the orthogonal complement TQ of the Hodge cycles. EndQ−HS (TQ ) must
be a totally real or CM field. Then a theorem of Zarhin implies M T (TQ ) must be a special
orthogonal group over F + a totally real field or a unitary group over F a CM field.
Fix a reference Weil cohomology H ∗ : P(k)op → GrE satisfying hard Lefschetz. (Eventually,
we’ll take k to be of characteristic 0.) Also fix a subfield E0 ⊆ E (we’ll have E0 = Q by default).
Let A•mot (X)E0 be the subset of elements of H ∗ (X)( 2• ) of the form
pX×Y
X∗ (α ^ ?β)
(9.8)
where α and β are algebraic cycles on X ×Y and ? is either ?L or ?H , say, associated to a product
polarization on X × Y . We want to know that we can compose correspondences. This works because
?X×Y can be related to ?X ⊗ ?Y ; this is cleaner for ?H than for ?L .
36
Suppose LX = ^ ηX , LY = ^ ηY ,and LX×Y = ^ ηX×Y , where ηX×Y = ηX ⊗ [Y ] +
[X] ⊗ ηY . Under the Künneth isomorphism, LX×Y →7 LX ⊗ 1 + 1 ⊗ LY . Also
X
o
→7
(i − dX − dY )πX×Y
X
X
i
(i − dX )πX
⊗
(j − dY )πYj
(9.9)
i≥0
and then Jacobson-Morosov implies we may complete to compatible sl2 -triples. Since Künneth
is an isomorphism of SL2 -representations, we find
0 −1
0 −1
0 −1
⊗
→7
1 0
1 0
1 0
(9.10)
Since the two matrices are ?H,X (−1)i and ?H,Y (−1)j , we deduce that ?H,,X×Y = (−1)ij ?H,X
⊗?H,Y in degree H i (X) ⊗ H j (Y ).
The basic calculation is:
Lemma 9.2.
(1) Amot (X)E0 is an E0 -subalgebra (with respect to ^) of H ∗ (X).
(X×Z)∗
X×Z
(2) PX∗
(Amot (X × Z)E0 ) ⊆ Amot (X)E0 , and pX
(Amot (X)E0 ) ⊆ Amot (X × Z)E0 .
Proof. For pushforward,
X×Z×Y
X×Z×Y
pX×Z
(· · · ).
X∗ (p(X×Z)∗ (α ^ ?X×Z×Y β)) = pX∗
(9.11)
Meanwhile, for pullback,
(X×Z)∗
pX
(X×Z×Y )∗
(α ^ ?β))
(9.12)
(X×Z×Y )∗
(α) ^ ?X×Z×Y (β ⊗ ?Z [Z])).
(9.13)
X×Y
(α ^ ?X×Y β)) = pX×Z×Y
(pX∗
(X×Z)∗ (pX×Y
±
= pX×Z×Y
(X×Z)∗ (pX×Y
Now, as for algebraic cycles, we define
r
X +r
Cmot
(X, Y ) = Admot
(X × Y )
(9.14)
with the composition law
r+s
Cmot
(X × Z)
s (Y × Z) × C r (X × Y )
Cmot
mot
(9.15)
β, α
×Z (X×Y ×Z)∗
pX×Y
(α)
(X×Z)∗ (pX×Y
^
(X×Y ×Z)∗
pY ×Z
(β))
0 (X, X)
(denoted simply by β ^ α). Then Cmot
E0 is a graded E0 -algebra, and we have formalism
∗
of f∗ , f , the projection formula, etc.
• (X, X) contains ? , ? , and π i .
Lemma 9.3. Cmot
L
H
X
37
Remark. For comparable Weil cohomologies, we get canonical identifications of the corresponding
spaces of motivated cycles.
(As a variant, one can restrict the auxiliary varieties (the Y appearing in the definition of
motivated cycles) to some full subcategory V of P(k) stable under ×, q, passing to connected
components, and containing P1 . We can then mimic the definition with V in place of P(k) as the
possible Y .)
We define the category of motivated motives over a field k of characteristic 0. Mmot
k , which we
write as Mk from now on, has:
0 (X, X), and m ∈ Z.
• Objects are (X, p, m) with X ∈ P(k), p an idempotent in Cmot
• The sets of morphisms are
n−m
Hom((X, p, m), (Y, q, n)) = qCmot
(X, Y )p.
(9.16)
Remark.
• If B(X) is true for every X, then Mmot
coincides with the previous category of
k
Grothendieck motives.
• As before, Mk is Q-linear and pseudo-abelian.
0 (X, X) is a finite-dimensional semisimple
Theorem 9.4 (analogue of Jannsen). For every X, Cmot
Q-algebra. Hence, Mk is abelian and semisimple.
Roughly, this is proven by defining anRanalogue of numerical equivalence. For x ∈ Amot (X), we
say x ∼#mot 0 if for every y ∈ Amot (X), X x.y = 0. As in Jannsen, we compare ∼mot and ∼#mot
via the map
X
X
Admot
(X × X) → Admot
(X × X)/ ∼#mot .
(9.17)
Translating Jannsen’s argument shows that the codomains are finite dimensional and semisimple.
But “we have B(X) and I(X) for motivated cycles”, so ∼mot =∼#mot . (Details may be found in §3
of André, IHES.)
i ∈ AdX (X × X), for k ⊆ C, we can modify
Mk is also a rigid ⊗-category, and because πX
mot
the commutativity constraint to obtain a neutral Tannakian category over Q, with fiber functor
∗ : M → Vect .
HB
Q
k
Here are some variants:
• We could restrict P(k) (or V) to some family of varieties W ⊆ P(k). We obtain a neutral
∗
W
Tannakian category over Q, MW
k with fiber functor HB , where Mk is the full subcategory
of Mk whose objects are isomorphic to subquotients of ⊕’s of ⊗’s of h(W ) and h(W )∨ for
W ∈ W. For example, W = {X} for some X.
∗ ) or G W = Aut⊗ (H • |
We get motivic Galois groups Gk = Aut⊗ (HB
). This lets us talk
B MW
k
k
about the motivic Galois group of a particular object M ∈ Mk , given by G M . This is an
analogue of M T (HB (M )).
• For E ⊇ Q, we can define the category Mk,E of motives modulo equivalence over k with
E-coefficients. (Objects may be considered as “E-modules in Mk ”.) There are corresponding
motivic Galois groups Gk,E .
38
Basic properties of Gk :
1. Gk is pro-algebraic, and in fact pro-reductive.
2. Gk splits over the maximal CM extension Qcm of Q. Here is one way to view this: for every E,
if M ∈ Mk,E , then M is isomorphic to N ⊗Ecm E for some N ∈ Mk,Ecm for Ecm the maximal
CM subfield of E.
This result follows from the existence of polarizations, and hence ultimately the Hodge index
theorem.
There are other manifestations of the principle that “arithmetic objects have CM coefficients”:
• The Frobenius eigenvalues of pure `-adic Galois representations are Weil numbers, in
particular in Qcm .
• Given an algebraic automorphic representation π, πf can be defined over Qcm , in cases
where we know enough to prove this. (π is unitary.)
10
Source of Motivated Cycles
There is a motivated analogue of the variational Hodge conjecture: suppose we have f : X → S
which is smooth and projective, with S smooth, projective, connected, reduced, and of finite type
over C. For s ∈ S, we get a specialization Xs → s.
Conjecture 10.1 (Variational Hodge Conjecture). If ξ ∈ H 0 (S, R2p f∗ Q(p)) is such that ξs ∈
2p
2p
(Xt )(p) is algebraic for every t ∈ S(C).
(Xs )(p) is algebraic for some s, then ξt ∈ HB
HB
Theorem 10.2 (André, building on Deligne, Blasins). The variational Hodge conjecture holds with
“motivated” in place of algebraic.
Here are the keys to the argument:
(1) MC is abelian.
(2) The theorem of the fixed part (from Hodge II).
We will review (2).
Theorem 10.3 (Theorem of the Fixed Part). Suppose we have f : X → S with f smooth and
projective, and S smooth and connected. Let X ,→ X be a smooth compactification. Then
H n (X, Q) → H 0 (S, Rn f∗ Q)
(10.1)
is surjective.
The relation to ”fixed part” is that for s ∈ S, we have a map H 0 (S, Rn f∗ Q) ,→ H n (Xs , Q) with
image H n (Xs , Q)π1 (S,s) .
39
“Proof ”. Consider the sequence of maps
a
c
b
H n (X, Q) −
→ H n (X, Q) →
− H 0 (S, Rn f∗ Q) ,−
→ H n (Xs , Q).
(10.2)
Observe:
(1) b is surjective. For b is an edge map in the Leray spectral sequence coming from X → S → ?,
but a theorem of Deligne implies that for f smooth and projective, Leray degenerates at E2 .
0,n
Hence H 0 (S, Rn f∗ Q), being equal to E20,n , must be E∞
.
(2) c ◦ b and c ◦ b ◦ a have the same image.
Assuming (2), since c is injective, b ◦ a has the same image as b, which is surjective. So it remains
to explain (2).
f
j
More generally, suppose we have Y −
→ X ,−
→ X with Y smooth and projective, X smooth, and
X a smooth compactification of X. Then in the sequence
H n (X, Q) → H n (X, Q) → H n (Y, Q)
(10.3)
the composition has the same image as the second map alone.
Here is why: each of these cohomology groups has an increasing weight filtration (Wi ) such
that GrW
i is a pure Hodge structure of weight i. For X, Y smooth and projective, Wn−1 = 0 while
Wn = H n (, Q) (so only GrW
i is nonzero). Meanwhile for X smooth but not necessarily projective,
Wn−1 = 0 while W2n = H n (X, Q), so GrW
i can be nonzero only for n ≤ i ≤ 2n.
Fact. Morphisms of mixed Hodge structures are strict for the weight filtration. That is, given
f : V1 → V2 of mixed Hodge structures, then for every i, f (Wi V1 ) = Wi V2 ∩ f (V1 ).
(As a consequence, mixed Hodge structures form an abelian category.)
Now we want im f ∗ = im(j ◦f )∗ . By strictness, it suffices to check that im(GrW f ∗ ) = im(GrW (j ◦
∗
f )∗ ). For m 6= n, GrW
m f = 0. In weight n, the result follows from the fact that
im j ∗ = Wn H n (X).
(10.4)
This follows from the definition of the weight filtration as (up to shift) the abutment of the
Leray filtration for the composite X ,→ X → ?.
Recall that we can arrange j : X ,→ X to be a S
smooth compactification with complement a
union of smooth divisors with normal crossings, D = i∈I Di . We then have
M
|Q|=n−a
M
H a (DQ , Q)(a − n) =
H a (X, QD (a − n))
|Q|=n−a
= H a (X, Rn−a j∗ Q)
n
=⇒ H (X, Q)
and the summand in the first expression is a pure Hodge structure of weight 2n − a.
40
(10.5)
Q
(10.6)
(10.7)
Fact. E3 = E∞ .
So, defining the weight filtration as the abutment of the spectral sequence,
H n (X, Q)
im(d2 = Gysin map)
Wn H n (X, Q) = E3n,0 =
(10.8)
so that im j ∗ is by definition Wn H n (X, Q).
Proof of motivated Variational Hodge Conjecture. Given f : X → S, we may assume that S is
connected and smooth. Suppose ξs is motivated; we wish to show ξt is motivated.
Choose a smooth compactification j : X ,→ X. By the theorem of the fixed part,
2p
HB
(X)(p)
H 0 (S, R2p f∗ Q(p))
js∗0
(10.9)
2p
HB
(Xs0 )(p)
j ∗0
s
with js∗0 having kernel independent of s0 . So in Ms0 , h2 (X)(p) −→
h2p (Xs0 )(p) has kernel inde0
∗
pendent of s (HB is exact). Let N be the quotient. Now we have the picture
h2p (X)(p)
jt∗
N
js∗
∼
∼
∗
js
im(js∗ )
h2p (Xs )(p)
∗
(10.10)
∗
jt
im(jt∗ )
∗
h2p (Xt )(p)
∼
→ im(jt∗ ), and so applying HB ,
By considering j t ◦ (j s )−1 , we get a map im(js∗ ) −
∼
H 2p (Xs )π1 (S,s) −
→ H 2p (Xt )π1 (S,t) .
(10.11)
We have ξs 7→ ξt , and ξs is motivated, so ξt is motivated. (Actually, we get a motivated cycle on
X with specializations ξs and ξt .)
As a consequence, the standard conjectures for MC would imply the variational Hodge conjecture.
Corollary 10.4. With f : X → S and s ∈ S as usual, let α ∈ H(Xs )⊗m ⊗ (H(Xs )∨ )⊗n be a
motivated cycle such that a finite index subgroup of π1 (S, s) acts trivially on α. Then all parallel
transports of α are motivated.
Proof. Apply the motivated variational Hodge conjecture after a finite étale base change S 0 → S.
41
Consequences of the motivated variational Hodge conjecture:
(1) Let A be an abelian variety over C. Then by Deligne-André, Hodge cycles on A are motivated.
More precisely, any Hodge cycle ξ on A has the form
A×Y
ξ = prA∗
(α ^ ?β)
(10.12)
for α and β algebraic on A × Y , where Y may be taken as a product of an abelian variety
and some abelian schemes over smooth projective curves. So the Hodge conjecture for abelian
varieties (not known!) would follow from the Lefschetz standard conjecture for Y an abelian
scheme over a smooth projective curve.
Corollary 10.5. For any abelian variety A over C,
GH
1 (A)
= M T (H 1 (A)).
(10.13)
(A priori, G M ⊇ M T (HB (M )) for every M .)
This follows from (1) because products of abelian varieties are abelian varieties.
Let (V, h) be a Q-Hodge structure. The Mumford-Tate group M T (V ) ⊆ GL(V ) is the Q-Zariski
closure of the image of h. (This is the smallest Q-subgroup of GL(V ) whose R-points contain h(S).)
Lemma 10.6. Consider all ⊗-constructions: for ν = {(ai , bi )} with ai , bi ∈ Z and 1 ≤ i ≤ t, let
ν
T =
t M
V
⊗ai
∨ ⊗bi
⊗ (V )
(10.14)
i=1
a Q-Hodge structure and representation of M T (V ). Then:
(1) A Q-subspace W ⊆ T ν is a sub-Hodge structure if and only if W is stabilized by M T (V ).
(2) t ∈ T ν is a Hodge class iff t is fixed by M T (V ).
Proof. (1) Take stab(W ) ⊆ GL(V ). Then W is a sub-Hodge structure iff S WR iff h factors
through stab(WR ) iff M T (V ) ⊆ stab(W ).
(2) Apply (1) to Q(1, t) ⊆ Q(0) ⊕ T ν where Q(0) has the trivial Hodge structure. t is fixed iff
Q(1, t) is M T (V )-stable iff (1) Q(1, t) is a sub-Hodge structure iff t is a Hodge cycle.
Corollary 10.7. The functor
Rep(M T (V )) → Q − HS → VectQ
(10.15)
is fully faithful and realizes M T (V ) as a Tannakian group for hV i⊗ ⊆ Q − HS (the Tannakian
subcategory generated by V ).
We’re interested in polarizable Hodge structures.
42
Lemma 10.8. Q − HSpol , a full subcategory of Q − HS, is semisimple.
So if V ∈ Q − HSpol , then M T (V ) is a connected reductive group.
Corollary 10.9. If V ∈ Q − HSpol , M T (V ) ⊆ GL(V ) is exactly the subgroup that fixes all Hodge
tensors.
Proof. This follows from the following general result (in characteristic 0): For H ,→ G ,→ GL(V )
with G reductive, let
H 0 = {g ∈ G|g fixes all tensors fixed by H}.
(10.16)
Then if H is reductive, then H 0 = H. (This applies to H = M T (V ) and G = GL(V ).)
To show this, for any H (reductive or not), there exists a representation W of G and a line
` ⊆ W such that H = stabG (`). If H is reductive, ` has a complement H-representation W 0 inside
W . Then W ⊗W ∨ ⊇ `⊗`∨ and H is the subgroup of elements fixing (any) generator of this line.
The general proof applies in the motivated setting as well:
Corollary 10.10. Let M ∈ Mk . Then using hM i⊗ , G M is exactly the subgroup of GL(HB (M ))
fixing all motivated cycles in all tensor constructions.
As a consequence, for M ∈ MC , M T (HB (M )) ⊆ G M , because motivated cycles are Hodge
cycles. The Hodge conjecture would imply equality.
1 (A)) = G H
Corollary 10.11. If A/C is an abelian variety, then M T (HB
1 (A)
.
Proof. For tensor powers,
1
1
HB
(A)⊗m ⊗ (HB
(A)∨ )⊗n ⊆ H m+n(2d−1) (Am+n )(some Tate twist)
(10.17)
and Am+n is an abelian variety.
A very hard (unknown except for easy cases like abelian varieties) conjecture, yet much weaker
than the Hodge conjecture, is that G M is connected.
The above corollary gives a supply of motivic Galois groups:
• The calculation of possible Mumford-Tate groups of abelian varieties, or more generally Mumford Tate groups of AV C = hAV’si⊗ , is essentially the Hodge-theoretic content of Deligne’s
Canonical Models paper (Corvallis).
• A coarser result is a (soft) general result of Zarhin giving an upper bound on the possible
Mumford-Tate groups and algebraic representations occurring in H k (X) for any X which is
smooth and projective. In the k = 1 case, Zarhin implies all simple factors of M T (H 1 (A)) are
classical groups, and any nontrivial representation of a simple factor is minuscule.
In particular, G2 (or any exceptional group) can’t occur as a Mumford-Tate group of an
abelian variety.
43
We might ask if G2 even arises as M T (V ) for some V ∈ Q − HSpol . This is at least a necessary
condition for it to be a motivic Galois group.
Proposition 10.12. A semisimple adjoint group M/Q is a Mumford-Tate group of a polarizable
Q-Hodge structure if and only if MR contains a compact maximal torus.
For example, we can’t get SLn or GLn for n > 2, or any Q-form of SO(3, 19). (However,
SO(3, 19) arises as a Mumford-Tate group of non-projective K3 surfaces.)
Proof. We’ll show that if M is simple and MR contains a compact maximal torus, then M is a
Mumford-Tate group. Let m = Lie(M ) and TR a compact maximal torus, fixed by some Cartan
involution θ of mR . That is, TR is fixed by some involution θ of mR such that if κ is the Killing form,
then the pairing
κθ (X, Y ) = −κ(X, θY )
(10.18)
is positive definite. Then we get a decomposition mR = hR ⊕ pR into θ = 1 and θ = −1 spaces,
and hR = Lie(TR ).
An h : S → MR → GL(mR ) will yield a polarizable Q-Hodge structure on m if and only if
Ad h(i) is a Cartan involution on mR . We’ll write down h: Choose a cocharacter λ : S 1 → TR (S 1
viewed as a real algebraic group) such that
(
0 α compact root
hλ, αi ≡
2 α noncompact root
(mod 4)
(10.19)
Here compact roots are those coming from hR and noncompact roots are those coming from pR .
Then extend λ to a weight zero (meaning trivial on Gm,R ) h : S → TR ⊆ MR .
Ad h(i) acts on the root space mγ by
ih,γi
(
1
γ compact
=
−1 γ noncompact
(10.20)
Now κ is negative definite on hR and positive definite on pR , so κ(, ) is a polarization on
(m, Ad h).
A “generic Mumford-Tate group” argument shows that many such choices exist.
Using this framework, it’s easy to check that G2 can’t arise as M T (H 1 (A)) for some abelian
variety A.
Consider the split form of G2 . Then hR = su(2) ⊕ su(2), with the simple compact roots being
the roots at angles a multiple of π2 .
Later, we’ll construct G2 as a motivic Galois group using the theory of rigid local systems,
originally due to Dettweiler-Reiter, via Katz. An alternative proof is due to Z. Yun, who also gets
E7 and E8 .
Here are a few more applications of the motivated variational Hodge conjecture:
44
2. The Kuga-Satake construction is “motivated”. That is, if X/C is a projective K3 surface,
there is an abelian variety KS(X) such that H 2 (X) ,→ H 1 (KS(X))⊗2 . The embedding, a
priori a morphism of Q-Hodge structures, is actually a morphism in MC . Thus we get the
Mumford-Tate conjecture for K3’s, etc.
3. Variation of motivic Galois groups in families: again the template is Hodge theory. Take a Qvariational Hodge structure, meaning Rn f∗ Q coming from f : X → S smooth and projective.
This has the holomorphically varying Hodge filtrations on fibers H n (Xs , C).
(In general, a holomorphic family of Hodge structures on S/C is a local system V on S and
a filtration by holomorphic sub-bundles F • ⊆ V ⊗Q OS .)
We would like to know how M T (Vs ) varies for s ∈ S(C).
For example, let E → Y be the universal elliptic curve over some modular curve and V =
R1 f∗ Q.
• At CM points s, M T (Vs ) is a torus of rank 2 over Q. These are dense in the classical
topology.
• But at non-CM points s, M T (Vs ) = GL2 .
Roughly, there is a “generic Mumford-Tate group” (GL2 in this case), and the Mumford-Tate
group drops on a countable union of closed analytic subvarieties.
Now we give the motivated analogue of a refinement of this assertion. We need some notion
of a family of motivated motives: A family of motives parameterized by a base S(C), with
S/k ⊆ C smooth and geometrically connected, consists of:
• Smooth projective S-schemes X and Y of relative dimensions dX and dY , equipped with
relatively ample line bundles LX/S and LY /S .
• Rational linear combinations Z1 and Z2 of closed integral S-subschemes of X ×S X ×S Y ,
flat over S, and such that the elements qs for s ∈ S(C) given by
Xs ×Xs ×Ys
qs = pr(X
[Z
]
^
?[Z
]
(10.21)
1,s
2,s
s ×Xs )∗
are in AdX (Xs × Xs ) and are idempotent.
• Fix j ∈ Z.
A family is then s 7→ (Xs , qs , j).
Theorem 10.13. (1) Let Exc be the subset of s ∈ S(C) such that G Ms does not contain the
image of a finite index subgroup of π1 (S(C), s) → GL(HB (Ms )). Then Exc is contained
in a countable union of closed analytic subvarieties of S(C) (not including S(C) itself ).
Alternatively, there exists a countable collection Vi of algebraic subvarieties Yi ⊆ Sk such
that Exc ⊆ Vi (C). (In Hodge theory, this continues to hold for arbitrary polarized Z-Hodge
structures, due to Cattani-Deligne-Kaplan.)
(2) There exists a local system (Gs )s∈S(C) of algebraic subgroups of GL(HB (Ms )) such that:
(i) G Ms ⊆ Gs for every s ∈ S(C).
/ Exc.
(ii) G Ms = Gs for every s ∈
(iii) Gs contains the image of a finite index subgroup of π1 (S(C), s).
(This is a purely topological input, regulating motivic data. Gs is the “generic MumfordTate group”.)
45
11
Rigid Local Systems
Let X/C be a smooth projective connected curve, S ⊆ X a finite set of points, and U = X \ X. For
now, we’ll work on the associated analytic spaces X an and U an .
A local system F of E-vector spaces on U an is a locally constant sheaf of E-vector spaces. That
is, for every x ∈ U , we associate an E-vector space Fx , and for every path γ : [0, 1] → U , we have
∼
an isomorphism ρ(γ) : Fγ(0) −
→ Fγ(1) , depending only on γ up to homotopy. A choice of basepoint
x gives an equivalence
∼
→ RepE (π1 (U, x)).
LocSys(U )E −
(11.1)
The case X = P1 is most interesting for our purposes. In this case, π1 (U an , x) is a free group on
|S| − 1 generators.
Basic question: given a local system F on U an , when does F “come from geometry”? For example,
does there exist a (smooth projective) family f : Y → U such that F ⊆ Rn f∗ E for some n?
One necessary condition for F to “come from geometry” is that the local monodromy at each
puncture must be quasi-unipotent. (The local monodromy is the image of a loop around the puncture, as an element of GL(Fx ). Quasi-unipotence means some power is unipotent, or equivalently
that all eigenvalues are roots of unity.)
Theorem 11.1 (Local Monodromy Theorem). Any polarizable Z-variation of Hodge structures over
the punctured disc ∆∗ has quasi-unipotent monodromy.
(For reference, see Schmid’s paper or Illusie in “p-adic periods volume”.)
Recall that a Q-variation of Hodge structures is a local system F on U an , and on E = F ⊗Q OU an
a filtration by holomorphic sub-bundles Fili (E) ⊆ E such that on each fiber Fx we get a Q-Hodge
structure on Fx with the Hodge filtration on Fx ⊗ C induced from the above filtration, satisfying
Griffiths transversality: there exists a connection ∇ : E → E ⊗ Ω1OU an with E ∇=0 = F, and mapping
Fili (E) into Fili−1 (E) ⊗ Ω1 .
Remark. Given f : Y → U a smooth projective family, the monodromy representation of Rn f∗ Q is
semisimple. (See Hodge II.)
There aren’t any known general sufficient conditions to come from geometry. But a guiding
philosophy (due to Simpson): rigid local systems should always come from geometry. Katz’s book
proves this is so for irreducible RLS’s on P1 \ S.
There are a few things we might mean by rigid:
An F ∈ LS(U an )E is physically rigid if for every G ∈ LS(U an )E such that for every s ∈ S,
F|∆∗s ∼
= G.
= G|∆∗s , we have F ∼
F is “physically semi-rigid” if there exist a finite number of local systems F1 , . . . , Fr such that
if G is locally isomorphic to F at all s ∈ S, then G is isomorphic to some Fi .
F is cohomologically rigid if, letting j : U ,→ X be the inclusion,
H 1 (X, j∗ End(F)) = 0.
(11.2)
This essentially means “there are no infinitesimal deformations with prescribed local monodromy”.
46
Physical rigidity is more intuitive, but hard to check. Meanwhile, cohomological rigidity is a
numerical condition and easier to check.
One way to motivate this definition: we have a “character variety” Hom(π1 (U an ), GLn (C)) which
is an affine variety. Then quotient by GLn acting by conjugation. This can be thought of as a moduli
space of local systems.
Locally, fix ρ : π1 (U an , x) → GLd (E). Define a functor on local artinian C-algebras with residue
field C,
Liftρ : ArtC → Set
(11.3)
where Liftρ (R) is the set of lifts of ρ to a map π1 → GLd (R). To determine the tangent space,
Liftρ (C[]) ∼
= Z 1 (π1 , Ad ρ).
(11.4)
For given ρe with ρe(g) = (1 + Ag )ρ(g), {g 7→ Ag } is a 1-cocycle.
To do a version with GLd -equivalence, let Def ρ be the category with objects (R, ρR : π1 →
GLd (R)) with ρR lifting ρ, and morphisms (R, ρR ) → (R0 , ρR0 ) are given by φ : R → R0 and
α ∈ GLd (R0 ) such that α (φ ◦ ρR ) = ρR0 . (Here α is conjugation by α.)
If we take ρ irreducible (which we will have to do eventually), reducing the condition α (φ ◦ ρR ) =
ρR0 modulo mR0 gives α ρ = ρ, so α ∈ C× (α is a scalar matrix). Then modifying α by an element
0
of (R0 )× , we get a conjugation relation α (φ ◦ ρR ) = ρR0 with α ∈ 1 + Md (mR0 ).
This motivates a definition of a more set-theoretic version of Def ρ where we uniformly throw
out scalar automorphisms. We define instead (for irreducible ρ) the functor
ArtC
R
Def ρ
Set
(11.5)
{lifts ρR of ρ}/((1 + Md (mR ))-equivalence)
The tangent space of Def ρ is then given by
Def ρ (C[]) ∼
= H 1 (π1 , Ad ρ).
(11.6)
This is the same correspondence as for Liftρ , noting that equivalence changes the result by a
1-coboundary.
We conclude that for ρ irreducible corresponding to F, H 1 (π1 (U ), End(F)) measures infinitesimal deformations of ρ. We want to consider deformations with prescribed local monodromy. Write
γs ∈ π1 (U ) for a generator of π1 (∆∗s ). Now consider the functor
ArtC
R
Def S
ρ
Set
(11.7)
{ρR lifting ρ, ∀ s ∈ S, ρ(γs ) ∼ ρR (γs )}/ ∼
where ∼ means up to conjugation by 1 + Md (mR ) in both cases. The tangent space is then
47
Def Sρ (C[]) = {g 7→ Ag |(1 + Aγs )ρ(γs ) ∼ ρ(γs )}.
(11.8)
The condition means that for every s, Aγs = Bs − ρ(γs ) Bs for some Bs ∈ Md (C). That is,
!
Def Sρ (C[])
res
1
= ker H (π1 (U ), Ad ρ) −−→
M
H
1
(π1 (∆∗s ), Ad ρ)
.
(11.9)
s∈S
Claim. The restriction map is identified with the Leray (for U ,→ X) edge map
H 1 (U, End(F)) → H 0 (X, R1 j∗ End(F)).
(11.10)
If we knew the claim, we’d get an identification of this kernel with H 1 (X, j∗ End(F)).
Sketch. Given some local system G on U , R1 j∗ G is the sheaf associated to the presheaf V 7→
H 1 (j −1 (V ), G) (and j −1 (V ) = U ∩ V ). If V ⊆ U , this is V 7→ H 1 (V, G). Covering V by simply
connected opens, local sections vanish, so under sheafification, we get R1 j∗ G|U = 0.
∼
In the neighborhood of a puncture, we get H 1 (∆∗s , G) −
→ R1 j∗ (∆s ), which glue to give global
sections
M
∼
H 1 (∆∗s , G) −
→ H 0 (X, R1 j∗ G).
(11.11)
s∈S
Cohomological rigidity is useful because it boils down to a numerical condition:
Lemma 11.2. The following are equivalent:
(1) F ∈ LS(U )irr is cohomologically rigid.
(2) χ(X, j∗ End(F)) = 2.
P
(3) 2 = χ(U )(dim F)2 + s∈S dim CentGLdim F (ρ(γs )).
Proof. The content is that for any G ∈ LS(U ),
χ(X, j∗ G) = χ(U )rank(G) +
X
dim((j∗ G)s ).
(11.12)
s∈S
((j∗ G)s gives the π1 (∆∗s )-invariants in the π1 (U )-representation.)
To see this, the Leray spectral sequence for j : U ,→ X formally implies
χ(U, G) =
X
(−1)b χ(X, Rb j∗ G)
(11.13)
b
= χ(X, j∗ G) −
X
= χ(X, j∗ G) −
X
dim H 1 (π1 (∆∗s ), G)
(11.14)
dim(Gx /(γs − 1)Gx ).
(11.15)
s∈S
s∈S
48
But we have an exact sequence
γs −1
0 → Gxγs =1 → Gx −−−→ Gx → Gx /(γs − 1)Gx → 0
(11.16)
implying dim H 1 (∆∗s , G) = dim Gxγs =1 = dim(j∗ G)s . We then get
χ(U, G) = χ(X, j∗ G) −
X
dim(j∗ G)s .
(11.17)
s
Finally, G is a local system on U , so χ(U, G) = χ(U )rank(G).
Remark. The lemma holds for lisse Q` -sheaves on U ,→ X, as long as they are tamely ramified (over
some algebraically closed field k in which ` is invertible). If there is wild ramification, there is a
more complicated correction term (Grothendieck-Ogg-Shafarevich).
For example, we have the matrix equation
1 −2
1 −2
1 0
=
2 −3
0 1
2 1
(11.18)
with each of the first two matrices conjugate to the Jordan block U (2) and the last matrix
conjugate to −U (2). This gives a local system on P1 \ {3 points}, which is cohomologically rigid:
χ(P1 \ 3)(rank(F))2 +
X
Cents = (−1)(2)2 + (2 + 2 + 2) = 2.
(11.19)
s
The local system is geometric; in fact, the matrices are the local monodromies of the Legendre
family E : y 2 = x(x − 1)(x − λ). Our F is then R1 f∗ C.
A more general class of equations are the hypergeometric local systems (on P1 \ {0, 1, ∞}).
Consider h0 , h1 , h∞ ∈ GLn (C) such that:
• h1 is a pseudo-reflection (rank(h1 − 1) − 1).
Q
• det(t − h∞ ) = (t − ai ).
Q
• det(t − h−1
(t − bj ).
0 )=
• h∞ h1 h0 = 1.
• ai 6= bj for every i, j.
This 3-tuple, regarded as a local system, is called a hypergeometric local system. Call H ≤
GLn (C) the subgroup generated by h0 , h1 , h∞ (called the “hypergeometric group”).
Lemma 11.3. Such an H is always irreducible (acting on Cn ).
Proof. Suppose not. Then H preserves some V1 ⊆ Cn and the quotient space V2 . h1 a pseudoreflection implies it must act trivially on one of V1 or V2 . So on one of V1 or V2 , h∞ = h−1
0 ,
contradicting the assumption on the ai and bj .
49
Hypergeometric local systems have an explicit matrix description:
Theorem 11.4 (Levelt). Let a1 , . . . , an , b1 , . . . , bn ∈ C× with ai 6= bj . Define the Ai and Bj by
Y
(t − ai ) = tn + A1 tn−1 + · · · + An
Y
(t − bi ) = tn + B1 tn−1 + · · · + Bn .
(11.20)
(11.21)
Now define the matrices

0 0 ···
1 0 · · ·


A = 0 1 · · ·
 .. .. . .
. .
.
0 0 ···

0 0 ···
1 0 · · ·


B = 0 1 · · ·
 .. .. . .
. .
.
0 0 ···

0 −An
0 −An−1 

0 −An−2 

..
.. 
.
. 
1 −A1

0 −Bn
0 −Bn−1 

0 −Bn−2 

..
.. 
.
. 
1 −B1
(11.22)
(11.23)
Then:
(I) Setting h∞ = A, h0 = B −1 , and h1 = A−1 B, we get a hypergeometric tuple with parameters
(ai ), (bj ).
(II) (Stronger than physical rigidity) Any hypergeometric local system with parameters (ai ), (bj ) is
GLn (C)-conjugate to the one given above.
Proof. (I) We only have to check that h1 is a pseudo-reflection. We have h1 = 1 = A−1 (B − A),
which has rank 1.
(II) Given such an h∞ , h1 , h0 , set A = h∞ T
and B = h−1
0 . Let W = ker(B −A). Then dim W = n−1
n−2 −j
since h1 is a pseudo-reflection. Then j=0
A W has dimension at least 1, so contains some
nonzero vector v. Now
v, Av, . . . , An−2 v ∈ ker(B − A)
(11.24)
We get Bv = Av, B 2 v = BAv = A2 v, and so on up to B n−1 v = · · · = An−1 v.
Claim. {v, Av, . . . , An−1 v} = {v, Bv, . . . , B n−1 v} is a basis of Cn .
Proof. By Cayley-Hamilton, A hv, . . . , An−1 vi (and also B acts similarly). By irreducibility,
this span must be Cn .
Now with respect to this basis, h∞ , h1 , h0 have the form we want.
50
Remark. The Jordan form of a companion block has a single Jordan block for each eigenvalue. This
implies such a local system F is cohomologically rigid:
X
dim Cent(hi ) = ((n − 1)2 + 1) + n + n = n2 + 2
(11.25)
0,1,∞
so χ(P1 , j∗ End(F)) = 2.
Questions we might ask:
1. What are the monodromy groups?
2. Are these local systems “geometric” (if the ai and bj are roots of unity)?
3. What is the relationship between physical and cohomological rigidity?
The first two answers are:
1. G2 cannot arise in this way (see Beukers-Heckman, where all possible monodromy groups of
hypergeometric local systems are computed. You get Sp, O, SL, and some finite groups).
2. Yes, by Katz’s theory.
Proposition 11.5. Let F ∈ LS(U )irr . Then if F is cohomologically rigid, F is physically rigid.
Remark. Here we take the analytic topology and a classical local system. But the identical result
holds with the same proof, for lisse `-adic sheaves on U ⊆ X over k algebraically closed with 1` ∈ k.
Proof. We are given that χ(X, j∗ End(F)) = 2. Let G ∈ LS(U ) have the same local monodromies
as F. Since End(F) and Hom(F, G) have the same local monodromies, the Euler characteristic
formula implies
2 = χ(X, j∗ End(F)) = χ(X, j∗ Hom(F, G)).
(11.26)
h0 (X, j∗ Hom(F, G)) + h2 (X, j∗ Hom(F, G)) ≥ 2.
(11.27)
In particular,
But H 2 (X, j∗ H) ∼
= Hc2 (U, H) for H lisse on U : take the exact sequence
0 → j! H → j∗ H → punctual → 0
(11.28)
and look at the long exact sequence in cohomology. So using Poincaré duality, we may rewrite
the above as
h0 (U, Hom(F, G)) + h0 (U, Hom(G, F)) ≥ 2.
(11.29)
Hence at least one of Hom(F, G) or Hom(G, F) has a nonzero global section. Since F is irreducible and F and G have the same rank, we get F ∼
= G as local systems on U .
51
Now, for the other direction, we will use a transcendental argument:
Proposition 11.6. If U ⊆ X = P1 and F ∈ LS(U an )irr is physically rigid, then F is cohomologically
rigid.
Proof. We need to check that χ(P1 , j∗ End(F)) = 2. We know that χ ≤ 2. So it suffices to check
that if F is physically rigid, then χ ≥ 2.
Let γ1 , . . . , γk be loops generating the monodromy, with γ1 · · · γk = 1. F is given by A1 , . . . , An
such that A1 · · · An = 1. Given G (D1 , . . . , Dn such that D1 · · · Dn = 1), if F and G have the same
local monodromies (for each i, there exists Bi such that Di = Bi Ai Bi−1 ), there exists C ∈ SLn (C)
such that Di = CAi C −1 . Now
χ(P1 , j∗ End(F)) = (2 − k)n2 +
k
X
dim Z(Ai ).
(11.30)
i=1
Q
π
Consider X = GLn (C)k −
→ SLn (C) by B1 , . . . , Bk 7→ ki=1 Bi Ai Bi−1 . Then the fiber at 1
corresponds to Q
local systems with the same local monodromy as F. This diagram is acted on by
G = SLn (C) × ki=1 Z(Ai ) where (C, Z1 , . . . , Zk ) acts by
B1 , . . . , Bk 7→ (CB1 Z1−1 , . . . , CBk Zk−1 )
A 7→ CAC
−1
(11.31)
.
(11.32)
Check that π is G-equivariant. So G acts on the fiber π −1 (1). F being physically rigid is equivalent to G acting transitive on π −1 (1). In particular,
dim G ≥ dim π −1 (1) ≥ dim X − (n2 − 1).
(11.33)
Then χ ≥ 2 follows.
Remark. The argument that physical rigidity implies cohomological rigidity works for H-rigid local
systems for H any reductive group. (Here π1 → H, and cohomological rigidity means H 1 (X, j∗ Lie(H)) =
0.) For general H, cohomological rigidity does not imply physical rigidity, however.
Remark. Given X/C, there exists an “analytification” functor
Dcb (X, Q` ) → Dcb (X an , Q` )
(11.34)
which is fully faithful, but not essentially surjective.
For example, take X = Gm . On X an , we have the Q` -local system given by the map π1 (X an ) →
taking a loop to 1` . This does not extend to the profinite completion π1ét (X) of π1 (X an ), so it
can’t arise from an étale local system.
×
Q`
∼
But given K ∈ Dcb (X an , C), for almost every `, we can choose i : C −
→ Q` such that i(K) is in
this essential image.
52
Theorem 11.7. Let k be a field. For variable X (separated and of finite type over k), we have a
triangulated category Dcb (X, Q` ), equipped with a “standard” t-structure such that
H0 : Dcb (X, Q` )♥ → {Q` -sheaves}
(11.35)
is an equivalence. Here the category {Q` -sheaves} is
1
lim {constructible OE -sheaves}
.
−→
$
E
E/Q
(11.36)
`
Varying X, we have the following adjoint functors: for f : X → Y , we have pairs (f ∗ , Rf∗ ),
(Rf! , f ! ), and ( ⊗L K, RHom(K, )).
Remark. For defining Dcb (X, Q` ):
• b means bounded, and c means constructible.
• Dcb (X, Q` ) = colimE/Q` (Dcb (X, OE ))E , but defining Dcb (X, OE ) takes work.
• There are three approaches one could take to define Dcb (X, O) for O = limr O$r :
←−
1. The pro-étale approach (Bhatt-Scholze).
2. The method used in Jacob’s class. Roughly, inverse limits are well-behaved for the stable
∞-category version of Dcb (X, O/$r ). In particular, the triangulated inverse limit comes
for free.
3. The classical approach, due to Deligne.
An example of what’s involved: replace Dcb (X, O/$r ) with the full-subcategory of very
well-behaved complexes (those that are quasi-isomorphic to bounded complexes of conb (X, O/$ r ). We want lim D b (X, O.$ r ) to
structive O/$r -flat sheaves). Call this Dctf
←− ctf
be naturally triangulated.
f
g
+1
A naive thing to try would be to say X −
→Y →
− Z −−→ is a distinguished triangle if for
fr
gr
+1
every r, Xr −→ Yr −→ Zr −−→ are distinguished triangles. Then we need to check the
triangulated axioms. For example, we need to verify
X
Y
Z
u
v
X0
Y0
+1
∃w
Z0
u[1]
(11.37)
+1
An obvious thing to do would be to let
Wr = {wr : Zr → Zr0 : diagram commutes}.
(11.38)
For the map Z → Z 0 in lim, we want an element of lim Wr .
←−
←−
Fact. (a) limr (nonempty finite sets) is nonempty.
←−
(b) For plenty of k, HomDcb (X,O/$r ) (Zr , Zr0 ) is finite. For example, this is true if k is
algebraically closed or finite, but not for k = Q.
53
12
Perverse Sheaves
We say that K ∈ Dcb (X, Q` ) is semi-perverse if for every i ∈ Z,
dim suppH−i K ≤ i.
(12.1)
K is perverse if both K and its Verdier dual D(K) are semi-perverse. As an instance of D, if
π : X → Spec K, we have
DK = RHom(K, π ! Q` ).
(12.2)
For example, if X is smooth of equidimension d, and F on X is a lisse Q` -sheaf, then F[d] is
perverse. In this case, π ! Q` = Q` [2d](d).
A general perverse sheaf is built out of lisse sheaves on smooth varieties.
Some general motivation for perverse sheaves:
1. They allow you to define intersection cohomology, IH ∗ (X), which even for (proper) singular
X satisfy Poincaré duality and purity (either in the `-adic or Hodge-theoretic sense).
2. We have a function-sheaf dictionary. Take X/Fq . Given a Q` -sheaf G on X, define, for every
m, a function
G
fm
X(Fqm )
x : Spec F
qm
→X
Q`
tr(f rm
(12.3)
|x∗ G).
This is identically 1 for G = Q` , but interesting for G = Rn f∗ Q` .
More generally, for K ∈ Dcb (X, Q` ), we may define
fK =
X
(−1)i f Hi (K) .
(12.4)
These functions interact nicely with sheaf-theoretic relations. For example:
(a) If K → L → M → K[1] is a distinguished triangle, then f K + f L = f M .
(b) f K⊗L = f K · f L .
(c) Given g : X → Y and K ∈ D(Y ), f g
∗K
= f K ◦ g.
(d) Given g : X → Y and K ∈ Dcb (X), then for y ∈ Y (Fqm ),
Rg! K
fm
(y) =
X
K
fm
(x).
(12.5)
x∈X(Fqm )
This follows from the Lefschetz trace formula.
If you have some classically understood operation on functions, we can use these sorts of
relations to mimic it at the level of sheaves.
The role of perverse sheaves is that we can recover a perverse sheaf from its functions:
54
Theorem 12.1. Two semisimple perverse sheaves K and L are isomorphic if and only if
K = f L for every m.
fm
m
Fact (hinted at in the theorem). Perv(X), the full subcategory of Dcb (X, Q` ) with objects the
perverse sheaves, is an abelian category with all objects having finite length.
What do perverse sheaves look like? That is, how do we produce more perverse sheaves starting
from the “lisse on smooth” case?
Theorem 12.2.
(1) If f : X → Y is affine, then Rf∗ preserves semi-perversity.
(2) If f : X → Y is quasi-finite, then Rf! (= f! ) preserves semi-perversity.
Corollary 12.3. If f is both affine and quasi-finite (for example an affine immersion), then both
f! and Rf∗ preserve perversity.
Proof. DRf∗ K ∼
= Rf! D(K).
j
Key construction (Intermediate extension): let Y ,−
→ X be a locally closed subvariety. This
factors into open and closed immersions Y ,→ Y ,→ X. For simplicity, take Y affine so j is affine
and quasi-finite. If K ∈ Perv(Y ), then both j! K and Rj∗ K are in Perv(X). There is a “forget
supports” map j! K → Rj∗ K. As Perv(X) is abelian, we may define
j!∗ K = im(j! K → Rj∗ K).
(12.6)
Properties:
1. j!∗ : Perv(Y ) → Perv(X) is fully faithful.
2. j!∗ D = Dj!∗ .
3. j!∗ preserves simple objects: it preserves injections and surjections.
This is the basic construction because:
Theorem 12.4. Any simple perverse sheaf K on X is of the form j!∗ (F[dimY ]) for some smooth
affine Y which is a locally closed subvariety of X, and some lisse sheaf F on Y .
L
Given K and X, where do F and Y come from? Look at supp Hi (K) = Y . Choose some
open dense Y ,→ Y such that the constructible sheaves Hi (K) become lisse when restricted to Y .
Then F = H−dY (K)|Y works.
(This was for every X/k separated of finite type.)
This is very concrete in the case where X is a smooth geometrically connected curve.
Fact. For U ,→ X open and F lisse on U , then j!∗ (F[1]) = j∗ F[1].
i
Remark. On X \ U ,−
→ X, for any sheaf G on X \ U ,
HomX (i∗ G, j∗ F) = Hom(j ∗ i∗ G, F) = 0
so “j∗ F has no punctual sections”.
55
(12.7)
More generally, K ∈ Dcb (X, Q` ) is perverse if and only if the following all hold:
• Hi (K) = 0 for i 6= −1, 0.
• H−1 (K) has no punctual sections. (This is j∗ F from before.)
• H0 (K) is punctual (supported at a finite set of points).
Hence simple perverse sheaves on X are either:
i
1. punctual: there exists a closed point x ,−
→ X such that K = i∗ (rank one local system on {x}),
or
j
2. there exists U ,−
→ X and F irreducible lisse on U such that K ∼
= j∗ F[1].
13
Middle Convolution
This is motivated by the sheaf-function dictionary.
Recall that hypergeometric local systems gave a basic supply of rigid local systems. For example, the Gauss hypergeometric function gave a rigid local system on P1 \ {3 points}, with local
monodromies of the form U (2), U (2), −U (2). The Gauss hypergeometric function has an integral
representation
1 1
F ( , , 1; λ) =
2 2
∞
Z
1
1
p
dx
x(1 − x)(λ − x)
(13.1)
(here the first three parameters determine the local monodromy). More generally, F (a, b, c; λ)
satisfies ∇F = 0, where
∇ = λ(1 − λ)
d
dλ
2
+ (c − (a + b + 1)λ)
d
− ab.
dλ
(13.2)
This has an integral representation
Z
F (a, b, c; λ) =
∞
xa−c (1 − x)c−b−1 (λ − x)−a dx.
(13.3)
1
R
This integral looks like f (x)g(λ − x) dx, an additive convolution. g(x) = x−a , a “function
associated to a Kummer sheaf”. For example, for a = 12 , [2] : Gm → Gm is étale, giving π1 (Gm ) →
×
Q` with image µ2 . The rank 1 sheaf corresponding to this representation is the associated Kummer
sheaf. Meanwhile f (x) = xa−c (1 − x)c−b−1 arises from a tensor product of (translated) Kummer
sheaves.
We want to express the U (2), U (2), −U (2) local system F in terms of simpler objects, namely
F ∼
1
1 . We have Lf g = Lf ⊗ Fg ; now we just need to understand what the
= L − 21 ? L
x
(1−x)− 2 x− 2
convolution ? is.
56
Let k be algebraically closed, G/k a connected smooth affine algebraic group over k, π : G×G →
G the multiplication. For K, L ∈ Dcb (G, Q` ) = D(G), we have two kinds of convolution:
K ?! L = Rπ! (K L)
(13.4)
K ?∗ L = Rπ∗ (K L).
(13.5)
However, even if K and L are perverse, the same is not necessarily true of K ?! L and K ?∗ L.
On the other hand, if K and L are semi-perverse, then so is K ?∗ L, because π is affine.
Suppose we had K ∈ Perv(G) such that for all L ∈ Perv(G), K ?! L and K ?∗ L are both
perverse. Then the middle convolution K ?mid L is the image (in the abelian category Perv(G)) of
the “forget supports” map K ?! L → K ?∗ L.
×
We implement this for G = A1 , and given χ : π1 (Gm ) → Q` nontrivial, we produce Lχ ∈
LS(Gm ), and take K = j∗ Lχ [1] for j : Gm ,→ A1 .
To see that ?mid j∗ Lχ [1] makes sense, we use:
Proposition 13.1. Take dim G = 1 affine, smooth, and connected. Suppose that the isomorphism
class in K ∈ Pervirr (G) is not translation invariant. Then ?! K and ?∗ K both preserve perversity.
Proof. First, the ?∗ statement follows from the ?! statement, for DK is also perverse and not
translation invariant. Then DK ?! L is perverse, hence so is its dual K ?∗ DL.
Next, if K, L ∈ Perv(G), then K ?! L is perverse if and only if it is semi-perverse. For its dual
DK ?∗ DL is semi-perverse, since π is affine. Now for K perverse, the following are equivalent:
(a) K ?! L is perverse for every L ∈ Perv(G).
(b) K ?! L is perverse for every irreducible L ∈ Perv(G).
To see this, recall that Perv(G) is an abelian category with all objects having finite length. So
given L as in (a), induct on the length of L: we may assume there exists a distinguished triangle
(M, L, N ) with M and N having lower length than L. Then (K ?! M, K ?! L, K ?! N ) is also a
distinguished triangle. The long exact sequence in cohomology gives
dim supp Hi (K ?! L) ≤ max dim supp (Hi (K ?! M ), Hi (K ?! N )) ≤ −i.
(13.6)
So we are reduced to showing that if K is perverse and not translation invariant and L is perverse
and irreducible, then K ?! L is semi-perverse. That is, we need to check that supp H0 (K ?! L) is
zero-dimensional and that Hi>0 (K ?! L) = 0.
Recall that a perverse irreducible on a curve was either punctual or a middle extension. If either
K or L is punctual, then K ?! L is just a translate of L or K, therefore perverse. So we can assume
there exists j : U ,→ G and lisse F and G on U such that K = j∗ F[1] and L = j∗ G[1].
The stalk of Hi (K ?! L) at a geometric point g ∈ G is
Hi (Rπ! (K L))|g = Ri π! (j∗ F[1] j∗ G[1])|g
∼
= H i+2 (π −1 (g), j∗ F ⊗ j∗ G|π−1 (g) ).
c
57
(13.7)
(13.8)
This clearly vanishes for i > 0, so we’re left to check that Hc2 is nonzero for at most a finite
number of g ∈ G. As π −1 (g) = {(gx, x−1 )}, a copy of G, our group is
Hc2 (G, trans∗g (j∗ F) ⊗ inv∗ (j∗ G)).
(13.9)
−1
Both trans∗g (j∗ F) and inv∗ (j∗ G) are lisse on Ug = trans−1
g (U ) ∩ inv (U ). So on Ug , by Poincaré
duality,
Hc2 (Ug , · · · ) ∼
= H 0 (Ug , trans∗g (F ∨ ) ⊗ inv∗ (G ∨ ))∨
(13.10)
Homπ1 (Ug ) (trans∗g F, inv∗ (G ∨ ))∨ .
(13.11)
=
∼
As these are both irreducible, either Hom = 0 or we get an isomorphism trans∗g F −
→ inv∗ G ∨ . If
Hom = 0 for all but finitely many points, we’ll be done. Suppose not. Then there exist infinitely many
∼
g for which trans∗g F −
→ inv∗ G ∨ . If this happens, since these g lie in the support of a constructible
sheaf on our curve G, it would have to happen for all g in some dense open V ⊆ G. If g0 ∈ V , so
for all g ∈ V , trans∗g F ∼
= trans∗g0 F, the isomorphism class of trans∗g0 F is invariant under g0−1 V . This
implies we get a subgroup of G containing g0−1 V , namely
{g ∈ G|trans∗g (trans∗g0 F) ∼
= trans∗g0 F}.
(13.12)
This subgroup must be all of G. So the isomorphism class of F, hence of the original K, is
translation-invariant, a contradiction.
We will apply this to G = A1 and K = j∗ Lχ [1] for χ a nontrivial character of π1 (Gm ).
We now state the main results, slightly specialized, about ?mid j∗ Lχ [1]. Take T` to be the full
subcategory of constructible Q` -sheaves F on A1 satisfying:
(1) F is an irreducible middle extension. That is, there exists j : U ,→ A1 with U open and dense,
∼
on which j∗ F is lisse, irreducible, and F −
→ j∗ j ∗ F.
(2) F is tame. That is, the π1 (U )-representation associated to j ∗ F is tamely ramified at the
punctures P1 \ U .
(3) F has at least two singularities in A1 . (If F has rank at least 2, then (1) and (2) imply (3).)
×
Fix χ : π1tame (Gm ) → Q` nontrivial, and let j0 : Gm ,→ A1 .
Theorem 13.2. For F ∈ T` , let
M Cχ (F) = (F[1] ?mid j0∗ Lχ [1])[−1].
(13.13)
Then:
(1) M Cχ : T` → T` . That is, M Cχ preserves irreducibility, yields non-punctual sheaves, and
punctures preserve the condition of being tamely ramified.
58
(2) We have composition laws M Cχ ◦ M Cρ ∼
= M Cχρ if χρ 6= 1. Also M Cχ ◦ M Cχ−1 ∼
= id.
(3) Recall that for F lisse on U , the index of rigidity is rig(F) = χ(P1 , j0∗ End(F)), so F is
cohomologically rigid if and only if rig(F) = 2.
Then for every F ∈ T` , rig(M Cχ (F)) = rig(F).
(4) The local monodromies of M Cχ (F) can be computed from those of F. (For reference, see
Dettweiler-Reiter.)
Remark (on the proof).
1. For preserving T` :
• Show that ?mid j0∗ Lχ [1] preserves the set of perverse irreducible K such that K ?! and K?∗ preserve perversity (we call this the set of K such that “K has P”. T` [1] lies
inside here.
There is a proof that works in any characteristic, but there is a pleasant approach in
characteristic p via the Fourier transform.
For any algebraically closed k and any X/k connected and separated of finite type,
define M E(X) ⊆ Perv(X) to be the full subcategory of Perv(X) with objects K such
that K ∼
= j!∗ j ∗ K for some j : U ,→ X with K lisse on U . We have an operation
M E(X) × M E(X)
j!∗ (F[d]), j!∗ (G[d])
⊗mid
M E(X)
(13.14)
j!∗ F ⊗ G[d]
If we take k to be of characteristic p 6= ∞, `, and X = A1 , it will turn out that the K
having P are in correspondence with M E(A1 ) via a Fourier transform map. This sends
?mid to ⊗mid .
×
A word about F T : Dcb (A1 ) → Dcb (A1 ): fix ψ : Fp → Q` incarnating an Artin-Schreier
sheaf Lψ on A1 . Then take
F Tψ,! (F) = Rpr2! (pr1∗ F ⊗ Lψ(xy) [1])
(13.15)
and define F Tψ,∗ similarly.
Fact. F Tψ,! = F Tψ,∗ . Write F Tψ for either of these.
As a consequence, F Tψ preserves perversity. For if K is perverse on A1 , then pr1∗ K[1] ⊗
Lψ(xy) is perverse on A1 ×A1 ; hence F Tψ,∗ preserves semi-perversity while F Tψ,! preserves
dual semi-perversity.
We also find that F T is involutive: F Tψ ◦ F Tψ(ψ?) ∼
= [−1]∗ (−1). In particular, F T is an
auto-equivalence of Perv(A1 ).
Now given K ∈ Perv(A1 ) having P, to check that K ?mid j0∗ Lχ [1] has P, we see that
F T (K) ⊗mid F T (j0∗ Lχ [1]) ∈ M E(A1 ). Then we check that if K is irreducible, so is
K ?mid j0∗ Lχ [1]. Indeed, we check that F T (K) ⊗mid j0∗ Lχ−1 [1] is irreducible. This is true
because j0∗ Lχ−1 [1] has generic rank 1, and tensoring with such an object is invertible.
Indeed, ⊗mid L ⊗mid DL is equivalent to the identity if L is lisse of rank 1.
• The complement of T` in the set of K having P is some explicit list; for example Q` [1]
appears. Then compute explicitly that the complementary set is preserved.
59
Now using the background theorem on properties of M Cχ : T` → T` , we can prove the classification theorem of tamely ramified cohomologically rigid local systems on P1 \ S over k algebraically
closed.
Other than M Cχ , we also use a simpler twisting operation: if L is lisse of rank 1 on U ⊆ P1 ,
M TL : T`,rank≥2 → T`,rank≥2 is defined by F 7→ j∗ (j ∗ F ⊗ j ∗ L). It’s easy to see that rig(M TL (F)) =
rig(F).
Now given an irreducible cohomologically rigid local system on P1 \ S, we aim to apply a series
of M Cχ ’s and M TL ’s to obtain a rank 1 object, which is then easy to understand. Both of these
operations are invertible, so we can then recover the original local system.
Theorem 13.3 (Katz). Suppose F ∈ T`,rank≥2 is lisse on P1 \ S = A1 \ D (for S = D ∪ ∞) and
cohomologically rigid. Then there exists a generic rank 1 L lisse on A1 \D and a nontrivial character
×
χ : π1 (Gm )tame → Q` such that G = M Cχ (M TL (F)) has strictly smaller rank than F.
Also, we can arrange L and Lχ to have “local monodromies contained in the local monodromies of
F”, so that the local monodromies of G are contained in those of F. In particular, if all eigenvalues
of local monodromy of F are in µN , then we can arrange the same for G.
Some bookkeeping: for each α ∈ D, we write
F|I(α)t ∼
=
h
i
Lχ(x−α) ⊗ Unip(α, χ, F)
M
(13.16)
×
χ:π1 (Gm )tame →Q`
where via x − α : A1 \ α → Gm , we identify π1 (Gm )tame with π1 (A1 \ α)tame ∼
= I(α)t . Similarly
at ∞, we write
F|I(∞)t ∼
=
h
i
Lχ ⊗ Unip(∞, χ, F)
M
(13.17)
×
χ:π1 (Gm )tame →Q`
with π1 (Gm )tame ∼
= I(∞)t .
For every s ∈ S, let ei (s, χ, F) be the number of Jordan blocks of length at least i in Unip(s, χ, F).
Now look at the rank formula: for all nontrivial χ,
rank M Cχ (F) = |D| rank F −
X
e1 (α, 1, F) − rank(F ⊗ Lχ )I(∞) .
(13.18)
α∈D
(Katz proves this via F T .) So to drop the rank, we want to arrange lots of 1’s as eigenvalues by
twisting, and the take M Cχ for χ maximizing (F ⊗ Lχ )I(∞) . Precisely:
1. For each α ∈ D, choose χα such that eN
1 (α, χα , F) is maximal. Then form L ∈ T` of rank 1 such
−1 . That is, take L =
that L|I(α)t ∼
χ
. Then M TL (F) has e1 (α, 1, M TL (F)) ≥
= α
α∈D Lχ−1
α (x−α)
e1 (α, ρ, M TL (F)) for every ρ.
2. Replace F by M TL (F) so F satisfies the result of the above. Choose χ such that dim(F ⊗
Lχ )I(∞) is maximal.
Claim. Any such χ is nontrivial, so the rank formula actually applies.
60
Suppose otherwise, so χ = 1. Then for every s ∈ D ∪ ∞, e1 (s, 1, F) ≥ e1 (s, ρ, F) for every ρ.
We find a contradiction in the Euler formula:
χ(P1 , j∗ End(F)) = (1 − |D|)(rank F)2 +
X
dim(End(F))I(s)
(13.19)
s∈S
{z
|
XXX
lemma
=
ei (s, χ, F)2
s
≤
XXX
s
=
χ
χ
X
}
(13.20)
i
e1 (s, 1, F)ei (s, χ, F)
(13.21)
i
e1 (s, 1, F) rank(F).
(13.22)
s
As F is cohomologically rigid, we find that
"
#
2 ≤ rank(F) (1 − |D|) rank(F) +
X
e1 (s, 1, F)
(13.23)
s
= χ(P1 , j∗ F)
(13.24)
which is nonpositive for F irreducible. We get a contradiction (precisely for F cohomologically
rigid). So χ is nontrivial.
Now to see that the rank actually drops, we have
rank M Cχ (F) = |D| rank(F) −
X
e1 (α, 1, F) − dim(F ⊗ Lχ )I(∞)
(13.25)
α∈D
= −χ(P1 , j∗ F) + rank(F) + e1 (∞, 1, F) − dim(F ⊗ Lχ )I(∞) .
(13.26)
So we just need to show that
χ(P1 , j∗ F) + dim(F ⊗ Lχ )I(∞) − e1 (∞, 1, F) > 0.
(13.27)
The same formula gives
"
#
χ(P1 , j∗ End(F)) ≤ rank(F) (1 − |D|) rank(F) +
X
e1 (s, χs , F)
(13.28)
e1 (α, 1, F) + e1 (∞, χ∞ , F)
(13.29)
s
where χs maximizes e1 at s. So χ = 2 implies
0 < (1 − |D|) rank(F) +
X
α∈D
= χ(j∗ F) − e1 (∞, 1, F) + e1 (∞, χ∞ , F)
(13.30)
and the last term is dim(F ⊗ Lχ )I(∞) because χ by definition is χ−1
∞ . So rank M Cχ (F) <
rank(F).
61
Because both of the steps in this process are reversible, we also find:
Corollary 13.4 (Deligne-Simpson problem, cohomologically rigid irreducible case). Given a tuple
of local monodromies at each s ∈ S, we can apply the algorithm to determine whether this tuple
actually arises as local monodromies of an irreducible cohomologically rigid local system.
As an exercise, the Dwork family
X0n+1 + · · · + Xnn+1 = (n + 1)tX0 · · · Xn
(13.31)
is a family π : X → P1 \ {∞, µn+1 } with Xt being the above. On each fiber Xt , we have an action
by the group
Y
n
o
Γ = (ζ0 , . . . , ζn ) ∈ µn+1
ζ
=
1
/(diagonal).
i
n+1
(13.32)
This gives a local system F = (Rn−1 π∗ Q` )Γ . F is not rigid. But there exists a rigid local
system G on P1 \ {0, 1, ∞} such that [n + 1]∗ G ∼
= F, where [n + 1] is the (n + 1)st power map
P1 \ {∞, µn+1 , 0} → P1 \ {0, 1, ∞}. G is actually hypergeometric, with local monodromies:
• At ∞, U (n).
• At 1, a pseudo-reflection.
• At 0, conjugate to

ζ
 ζ2




..
.
ζn




(13.33)
where ζ is a primitive (n + 1)st root of unity.
An exercise is to, just starting with the local monodromies, apply Katz’s algorithm to reduce
this to rank 1.
14
Construction of G2 Local Systems
Let k be algebraically closed of characteristic not equal to ` (or 2?).
×
Theorem 14.1. (1) Fix α1 6= α2 ∈ A1 (k). Let ϕ, η”π1t (Gm ) → Q` such that ϕ, η, ϕη 2 , ηϕ2 , ϕη −1
are all not equal to −1. Then there exists an irreducible cohomologically rigid local system
F = F(ϕ, η) ∈ T` of rank 7 with the following local monodromies:
• At α1 , −1⊕4 ⊕ 1⊕3 .
• At α2 , U (3) ⊕ U (2) ⊕ U (2).
• At ∞, any of the following:
62
–
–
–
–
–
U (7), for ϕ = η = 1.
ϕU (3) ⊕ ϕ−1 U (3) ⊕ 1, for ϕ = η 6= 1, ϕ3 = 1.
ϕU (2) ⊕ ϕ−1 U (2) ⊕ ϕ2 ⊕ ϕ−2 ⊕ 1, for ϕ = η, ϕ4 6= 1, ϕ6 6= 1.
ϕU (2) ⊕ ϕ−1 U (2) ⊕ U (3), for ϕ = η −1 , ϕ4 6= 1.
ϕ ⊕ η ⊕ ϕη ⊕ ϕη −1 ⊕ η −1 ⊕ ϕ−1 ⊕ 1, when these 7 characters are all distinct.
In each case, the monodromy group (the Zariski closure of the image of the monodromy
representation) is G2 ,→ GL7 .
• Let F ∈ T` be cohomologically rigid, ramified at ∞, with monodromy group G2 . Then F
is ramified at exactly two points of A1 , say α1 , α2 . Up to permuting {α1 , α2 , ∞}, F is
conjugate to one of the local systems in (1).
Proof. We’ll first prove (2). Suppose F is lisse exactly on A1 \D. Cohomological rigidity is equivalent
to having
X
29(|D| + 1) ≥
dim Cent(F|I(s) )
(14.1)
s∈D∪∞
= 49(|D| − 1) + 2
(14.2)
sow 20|D| ≤ 76, implying |D| ≤ 3. Suppose |D| = 3. Then 100 equals the sum of four centralizer
dimensions. The only possibilities for the centralizer dimensions are (25, 25, 25, 25), (29, 29, 29,
13), and (29, 29, 25, 17). But we also have the following necessary irreducibility criterion that
0 ≥ χ(P1 , j∗ F)
= 7(1 − |D|) +
(14.3)
X
e1 (s, 1, F)
(14.4)
s∈D∪∞
X
e1 (s, 1, F) ≤ 14.
(14.5)
s
We can show that this rules out all of the above possibilities. For example, in the (25, 25,
25, 25) case, the Jordan form is −1⊕4 ⊕ 1⊕3 . If we twist by a character with local monodromies
−1, −1, −1 at the finite points (and then −1 at ∞), we get a new irreducible local system with local
monodromies 1⊕4 ⊕ −1⊕3 at each puncture. This violates the irreducibility criterion.
We see that |D| = 2. In this case, cohomological rigidity is equivalent to the sum of the three
centralizer dimensions being 51. Look in the table for the possible triples; they are (29, 13, 9), (29,
11, 11), (25, 19,
P 7), (25, 17, 9), (25, 13, 13), (19, 19, 13), and (17, 17, 17). But the irreducibility
criterion says s e1 (s, 1, F) ≤ 7, which immediately excludes the first, fourth, fifth, and seventh
cases.
Since the monodromy group is assumed to be all of G2 (not just contained in G2 ), the 14dimensional adjoint representation π1 → G2 → GL(g2 ) also must be irreducible. Now the necessary
irreducibility criterion requires
X
dim CentG2 ≤ 14.
α1 ,α2 ,∞
63
(14.6)
This rules out the remaining cases, except for (25, 19, 7), and in this case the G2 centralizers
must be (6, 6, 2) instead of (6, 8, 2).
This implies the only possibilities are as follows: the 25/6 must come from −1⊕4 ⊕ 1⊕3 , the
19/6 must come from U (3) ⊕ U (2) ⊕ U (2), and the 7/2 has several possibilities. (2) of the theorem
is a slight refinement of this, checking which of these cases can arise (and getting the condition of
ϕ, η, . . . 6= −1). These cases are ruled out by running Katz’s algorithm and finding contradictions.
Part (1) is then proven by taking the leftover cases and running Katz.
For example, consider the case where the 7/2 monodromy is U (7) at ∞, and the 25/6 and 19/6
monodromies are at α1 and α2 . Given characters χ1 , χ2 of π1t (Gm ), write L(χ1 , χ2 ) for the rank 1
local system with monodromies χ1 at α1 and χ2 at α2 . We get
⊕L(−1,1)
U (7)// − 1⊕4 , 1⊕3 //U (3), U (2), U (2) −−−−−−→ −U (7)//1⊕4 , −1⊕3 //U (3), U (2), U (2)
(14.7)
M C−1 (?)
−−−−−−→ U (6)//U (2)⊕3 // − U (2), −1, −1, −1, −1
L(1,−1)
−−−−−→ · · ·
(14.8)
(14.9)
M C−1
−−−−→ U (5)// − 1, −1, −1, 1, 1// − 1, U (2), U (2)
(14.10)
..
.
(14.11)
M C−1
−U (2)//U (2)//U (2) −−−−→ 1// − 1// − 1
(14.12)
and such a rank 1 local system exists since 1 · −1 · −1 = 1. Now to prove that the original F
exists, start with this rank 1 output and run this whole procedure in reverse. The other possible
regular local monodromies are dealt with similarly.
(?) : rank = 7 − 4 + 7 − 3 − 1 = 6. See the opposite side of the reference sheet.
The last thing to do is to prove that the monodromy group is actually G2 in these cases.
We’ll consider the U (7) case again. First, our ρ : π1t → GL7 is orthogonal: the dual representation
g 7→ t ρ(g)−1 has GL7 -conjugate local monodromies to those of ρ. Since ρ is cohomologically rigid
(hence physically rigid!), this implies ρ ∼
= ρ∨ . Since the rank is odd, this can’t be symplectic, so
must be orthogonal.
So we may assume ρ has image in O7 (Q` ).
Fact. An irreducible subgroup G of O7 (Q` ) lies inside an O7 -conjugate of G2 ⊆ O7 if and only if
7
(Λ3 Q` )G 6= 0.
So for U = A1 \ {α1 , α2 }, we need H 0 (U, Λ3 F) 6= 0. That is, H 0 (P1 , j∗ Λ3 F) 6= 0. By Poincaré
duality, this is equivalent to H 2 (P1 , j∗ Λ3 F) 6= 0. Now we compute the Euler characteristic:
χ(P1 , j∗ Λ3 F) = − rank(Λ3 F) +
= −25 +
X
dim(Λ3 F)I(s)
s=α1 ,α2 ,∞
7
dim(Λ3 Q` )U (7) + dim(Λ3 Q` )U (3),U (3),U (2)
(14.13)
+ dim(Λ3 Q` )−1
⊕4 ,1⊕3
.
(14.14)
For the semisimple case, the number of 1 eigenvalues on Λ3 is 42 · 3 + 1 = 19. For the regular
case, U (7) is the image of U (2) under Sym6 : SL2 → SL7 . We compute the plethysm
64
Λ3 ◦ Sym6 = Sym12 ⊕ Sym8 ⊕ Sym6 ⊕ Sym4 ⊕ 1.
(14.15)
7
By SL2 -theory, dim(Λ3 Q` )U (7) is the number of irreducible constituents of this representation
over SL2 , equal to 5.
For the final case, our map SL2 → SL7 taking U (2) to U (3), U (2), U (2) is Sym2 ⊕ std ⊕ std.
Expand Λ3 (Sym2 ⊕ std ⊕ std) and count the number of irreducible constituents; there are 13.
We get
χ(P1 , j∗ Λ3 F) = −35 + 19 + 5 + 13 = 2
(14.16)
which is positive, so H 0 (U, Λ3 F) 6= 0.
This implies our ρ has image contained in G2 .
Fact (Dynkin). An irreducible subgroup of G2 containing a regular unipotent is either SL2 or G2 .
The U (3), U (2), U (2) class rules out the possibility of being SL2 .
As partial justification for the table’s results, we’ll explain why G2 contains elements having
GL7 -Jordan forms U (7); U (3), U (2), U (2); and −1⊕4 ⊕ 1⊕3 .
Let ∆ = {α1 , α2 } be a basis of simple roots for G2 and X 0 (T ) = Zα1 ⊕ Zα2 the root lattice.
Take β1 , β2 a dual basis so X0 (T ) = Zβ1 ⊕ Zβ2 . The fundamental weights are $1 = 2α1 + α2 and
$2 = 3α1 + 2α2 . The 7-dimensional (quasi-minuscule, meaning having the weights of V$1 , so the
Weyl orbit of $1 is 0) is V$1 . That is, the weights of V$1 are {±(2α1 + α2 ), ±(α1 + α2 ), ±α1 , 0}.
The coroots are α1∨ = 2β1 − 3β2 and α2∨ = −β1 + 2β2 , computed using h$1 , αj∨ i = δij .
Obtaining the semisimple element is easy. Take hβ, (−1), weights of V$1 i = (−1)hβi ,weightsi =
−1⊕4 , 1⊕3 .
For the U (7) element, recall that for any semisimple Lie algebra g, there is a unique maximal
nilpotent orbit which is open and dense on the set of nilpotents in g. Call an element in this orbit
regular nilpotent. An obvious guess in G2 to produce a U (7) would be to take a regular unipotent.
But note that a map H1 → H2 (even “irreducible”) need not take regular unipotents to regular
unipotents.
To construct regular nilpotents, take a basis ∆ of simple roots. For each α ∈ ∆, let Xα ∈ gα \ 0
and form
P the sl2 -triple {Xα , Hα , Yα } provided by basic structure theory. A regular nilpotent is then
X = α∈∆ Xα . We can put X inside an sl2 -triple, called the “principal sl2 ”, as follows:
There existsP
H ∈ t = Lie(T ) such that
P for every α ∈ ∆, α(H) = 2. Then there exist unique aα ’s
such that H = α∈∆ aα Hα . Set Y = α∈∆ aα Yα . Then {X, H, Y } is an sl2 -triple.
ρ$1
0
In our setup, we have a map sl2 → g2 −−→
GL7 with 10 −1
7→ H. We compute
hH, weights of ρ$1 i = 6, 4, 2, 0, −2, −4, −6.
(14.17)
So the composite sl2 → g2 → gl7 is Sym6 , hence X maps to a regular nilpotent in gl7 which
lands in g2 .
Now consider U (3), U (2), U (2). To any subset Θ ⊆ ∆, associate the parabolic pΘ that contains
gα , g−α for every α ∈ Θ. For g2 , the only possibilities are g2 , the Borel b, and two intermediate
65
parabolics p{α1 } and p{α2 } . Each p{α} has Levi subgroup l{α} = t ⊕ gα ⊕ g−α (the semisimple part
ρα
ρ$
1
is sl2 ). Take a regular nilpotent in sl2 −→ g2 −−→
gl7 for each α. We want to compute the image of
Xα in gl7 .
We have to compute hHα , γi = hγ, α∨ i for weights γ. Check that hα2 , α1∨ i = 3 so
hHα1 , weightsi = hweights, α1∨ i = ±1, ±1, ±2, 0.
(14.18)
The composite sl2 → gl7 is then Sym2 ⊕ std ⊕ std. (If we did Xα2 instead, we would get
U (2), U (2), 1⊕3 .)
15
Universal Rigid Local Systems
We would like to construct “universal rigid local systems with given local monodromies”.
Let k be an algebraically closed field of characteristic not equal to `, and α1 , . . . , αn ∈ A1 (k).
Fix an “order of quasi-unipotence” N ∈ Z6=0 , not divisible by the characteristic of k. Let Fk ∈ T` be
cohomologically rigid and lisse on A1 \ {α1 , . . . , αn }, with eigenvalues in µN . For bookkeeping, fix ζ
a primitive N th root of unity in k.
We will show how to produce a local system over an “arithmetic configuration space” whose
geometric fibers over k 0 are cohomologically rigid objects of T` (k 0 ), one of which is the original Fk .
The arithmetic part arises from working over RN,` = Z[µN , N1` ], with a fixed nonzero map to k.
We have a configuration space over RN,` : let
"
SN,n,`
#
1
= RN,` [T1 , . . . , Tn ] Q
.
i6=j (Ti − Tj )
(15.1)
Our “universal rigid local system” will live on A1SN,n,` \ {T1 , . . . , Tn }, where A1 has coordinate
X. This space is then
1
Spec RN,` [T1 , . . . , TN , X] Q
(Ti − Tj )
1
Q
(X − Ti )
.
(15.2)
If we have a local system her, we can specialize to k via RN,` → k and Ti 7→ αi to get a local
system on A1 \ {α1 , . . . , αn } over k.
Let j : A1SN,n,` \ {T1 , . . . , Tn } ,→ A1SN,n,` be the inclusion.
Theorem 15.1. With the same setup as before (in particular, given Fk ), fix RN,` ,→ Q` inducing
some map Eλ = Frac(RN,` )λ ,→ Q` . Then:
(1) There exists a lisse Eλ -sheaf F on A1SN,n,` \{T1 , . . . , Tn } which, after specializing along SN,n,` →
k, recovers Fk |A1 \{α1 ,...,αn } .
(2) The restriction of j∗ F to any geometric fiber is a cohomologically rigid object of the corresponding T` . That is, specialization preserves tameness, irreducibility, and index of rigidity,
“with the same local monodromies as Fk ”.
66
(3) F is pure (of some integer weight) and all polynomials of the form char(Frob) lie in Z[ζN ].
(4) For any other prime `0 and λ0 : Z[ζN ] ,→ Q` , there exists a lisse Eλ0 -sheaf Fλ0 on A1S
N,n,`0
{T1 , . . . , Tn } satisfying (the previous parts of the proposition and) for every ψ :
{T1 , . . . , Tn } → Fq ,
char(Frobψ |Fλ0 ) = char(Frobψ |F)
A1S
N,n,`,`0
\
\
(15.3)
in Z[ζN ]. That is, we get a compatible system of `-adic representations.
“Proof ”. For now, we assume there is a well-behaved notion of middle convolution with parameters
(that is, one on A1R for some reasonable ring R, such as SN,n,` which we use here).
×
Induct on rank Fk .NThe base case is rank 1. Let χi : I(αi ) → Q` be the local monodromy
n
character. Then Fk ∼
= i=1 Lχi (X−αi ) .
Interpret Lχi (X−αi ) as {y N = x − αi } embedded in Gm × A1 \ {αi }, with Galois group µN (k)
acting via (x, y) 7→ (x, ζy). From the fixed RN,` → k, we get µN (SN,n,` ) = µN (RN,` ) = µN (k), so
view χ instead as a character of the Galois group of {y N = X − Ti } over A1SN,n,` \ {Ti }. We get
N
a lisse sheaf Lχi (X−Ti ) on A1SN,n,` \ {Ti }. Now F = ni=1 Lχi (X−Ti ) satisfies the conditions of the
theorem in rank 1.
Now for the inductive step, suppose rank Fk ≥ 2. By the main theorem (Katz’s algorithm), we
×
can find a lisse sheaf Lk of rank 1 on A1 \ {α1 , . . . , αn } and a nontrivial χ : π1 ((Gm )k ) → Q` such
that Gk = M Cχ (M TLk (Fk )) has rank less than rank Fk . Spread out Lk to L as before. By induction,
we can spread out Gk to G on A1SN,n,` \ {T1 , . . . , Tn }.
Now invoke “MC with parameters”. Let j0 : (Gm )SN,n,` ,→ A1SN,n,` . Then the F we are looking
for is
F = L−1 ⊗ j ∗ j∗ G[1] ?mid j0∗ Lχ [1] .
(15.4)
Now we want to make sense of ?mid on A1R in a way that commutes with base change on R, and
for R = k specializes to the old construction.
15.1
Middle Convolution With Parameters
Q
Let R be a normal domain, of finite type over Z, and the divisor D ⊆ A1RQgiven by (X − ri ) = 0,
where ri − rj ∈ R× for i 6= j. We may also consider another D0 given by (X − ri0 ) = 0.
For us, we consider
1
1
R = SN,n,` = Spec Z ζN ,
[T1 , . . . , Tn ] Q
N`
(Ti − Tj )
and D :
Q
(X − Ti ) = 0, and D0 = {0} (meaning X = 0).
67
(15.5)
We will consider middle convolution
as a map LS(A1 \ D) × LS(A1 \ D0 ) → LS(A1 \ (D ? D0 ))
Q
where D ?D0 is the vanishing of i,j (X −(ri +rj0 )). In our case, D ?D0 = D. Define A(2) (dependent
on D and D0 as)
1
1
1
A(2) = Spec R X1 , X2 ,
,
,
gD (X1 ) gD?D0 (X2 ) gD0 (X2 − X1 )
(15.6)
where gD is the polynomial defining D, etc. A(2) comes equipped with maps
A(2)
pr2
A1 \ D 0
A1R \ D
(15.7)
d
pr1
A1R \ (D ? D0 )
where d(X1 , X2 ) = X2 − X1 . Consider j : A(2) ,→ P1 × (A1 \ (D ? D0 )). Then pr2 : P1 × (A1 \
(D ? D0 )) → A1 \ (D ? D0 ) is proper and smooth.
Now for F ∈ LS(A1R \ D) and F 0 ∈ LS(A1R \ D0 ), we define
F ?naive F 0 = R1 pr2! (pr1∗ (F) ⊗ d∗ F 0 )
0
F ?mid F = R
1
pr2∗ (j∗ (pr1∗ F
∗
(15.8)
0
⊗ d F )).
(15.9)
Proposition 15.2. With the setup as before, either assume that everything is tame or that R has
generic point of characteristic zero. Then:
(a) F ?naive F 0 and F ?mid F 0 are lisse (and tame).
(b) Assume that F and F 0 are pure of weights w and w0 . Then F ?naive F 0 is mixed of weights at
most w + w0 + 1, while F ?mid F 0 is pure of weight w + w0 + 1. In fact,
0
F ?mid F 0 ∼
= GrW
w+w0 +1 (F ?naive F ).
(15.10)
(c) Assume that F or F 0 is geometrically irreducible and nonconstant. Then for G = pr1∗ F ⊗ d∗ F 0 ,
we have Ri pr2! (G) = 0 unless i = 1, and Ri pr2∗ (j∗ G) = 0 unless i = 1.
Parts of proof. (a) Roughly, tame means we can think topologically. Take D0 = {0}. Then A(2)
is the complement of hyperplanes X1 = ri , X2 = ri , and X1 = X2 . R1 pr2∗ (j∗ G) is the
sheaf associated to U 7→ H i (pr−1
). The projection pr2 : (A(2), (P1 × (A1 \
2 (U ), j∗ G|pr−1
2 U
D)) \ A(2)) → U is trivial even with respect to the stratification, so the above group is
H i (pr2−1 (u0 ) × U, jU ∗ (· · · |pr−1 (u0 ) ) (Q` )U )U , so you get a local system.
2
(For a formal proof, see Katz, Sommes Expon., §4.7.)
(b) To show F ?naive F 0 is mixed, recall Weil II.
A sheaf F on a scheme Y of finite type over Z is pure of weight w ∈ Z if for every closed
point s : Spec Fq → Y , s∗ F is pure of weight w in the familiar sense that the eigenvalues of
w
f rq have absolute value q 2 under every complex embedding. F on Y is mixed if it admits an
increasing filtration by subsheaves such that Gri F are all pure.
68
Theorem 15.3 (Weil II). Given f : X → Y a morphism of schemes over finite type over Z,
with ` invertible on X and Y , if F is mixed of weights at most w on X, then Ri f! F is mixed
of weights at most w + i.
Now it’s immediate that the naive convolution is mixed with the specified upper bound on
weights.
To see that F ?mid F 0 is pure, the analogous statement for curves over Fq is
Theorem 15.4 (Weil II). Let j : U ,→ X for X a smooth curve, and F lisse on U and pure
of weight 1. Then H 1 (X, j∗ F) is pure of weight w + 1.
This implies the claim about middle convolution. Finally, we claim that
0
F ?mid F 0 = GrW
w+w0 +1 (F ?naive F ).
(15.11)
0 → j! G → j∗ G → i∗ i∗ j∗ G → 0.
(15.12)
To see this, we have
Taking the long exact sequence for Rpr2∗ , we get
R0 pr2∗ (i∗ i∗ j∗ G) → F ?naive F 0 → F ?mid F 0 → R1 pr2∗ (i∗ i∗ j∗ G) = 0.
(15.13)
Then it’s enough to know that R0 pr2∗ (i∗ i∗ j∗ G) is mixed of weights w + w0 . This follows for
1.8.9 of Weil II (j∗ takes weights at most r to weights at most r).
(c) We’ll show the !-version holds. That is, Ri pr2! (G) = 0 for i 6= 1. We check on geometric
fibers. Lefschetz affine implies it suffices to show for i = 2. In this case, the s-fiber is Hc2 (A1 \
D, G|s-f iber ). G is lisse on A1 \ (D ∪ {s}), and Hc2 is the coinvariants in monodromy, meaning
Gπ1 (A1 \(D∪s)) . In our case, pr1∗ F is unramified along X1 = X2 . But d∗ Lχ is ramified along
X1 = X2 . So local monodromy under X2 7→ s has ramification of the Kummer sheaf (χ 6= 1).
This does recover the old definition of convolution when R is an algebraically closed field (still in
the tame case). (We only wanted finite type over Z to get the weight filtrations.) There are two steps
to show this: the old convolution of F lisse on A1 \D and Lχ on A1 \0 was (jD∗ F[1]?mid j0∗ L[1])[−1],
where K ?mid Kχ = im(K ?! Kχ → K ?∗ Kχ ) ∈ Perv(A1 ). If we take
A1 × A1
j
P1 × A1
pr2
∞ × A1
pr2
(15.14)
A1
then Rpr2! (pr1∗ K ⊗ d∗ Kχ ) = K ?! Kχ , and K ?∗ Kχ has a similar description.
Lemma 15.5.
K ?mid Kχ = Rpr∗2 (j!∗ (pr1∗ K ⊗ d∗ Kχ )).
69
(15.15)
From here, show that Katz §2.8 or 2.9 can replace j!∗ with j∗ .
Lemma 15.6. Given the diagram
U
j
X
f
D
i
(15.16)
f
f |D
S
with D = X \ U , j open affine, f proper, f |D finite, and S separated of finite type over k, an
algebraically closed field of characteristic not equal to `, suppose K ∈ Perv(U ) is such that both
Rf! K and Rf∗ K are perverse. Then im(Rf! K → Rf∗ K) in Perv(S) is Rf ∗ (j!∗ K).
Proof. We have two short exact sequences in Perv(X):
0 → ker → j! K → j!∗ K → 0
(15.17)
0 → j!∗ K → Rj∗ K → coker → 0
(15.18)
and
with ker and coker supported on D, because j! K, Rj∗ K, and j!∗ K all extend K. Now applying
Rf ∗ gives two distinguished triangles on S:
(Rf ∗ ker, Rf ∗ j! K , Rf ∗ j!∗ K)
| {z }
(15.19)
(Rf ∗ j!∗ K, Rf ∗ Rj∗ K , Rf ∗ coker).
| {z }
(15.20)
Rf! K
and
Rf∗ K
Everything except possibly Rf ∗ j!∗ are perverse, and this object is as well by the long exact
sequence on H i (or p H i ). Hence these distinguished triangles are short exact sequences in Perv(S).
Now splice these exact sequences together to get the desired claim.
This finalizes the description of the middle convolution algorithm in the “universal context”: it
produces the local systems on A1SN,n,` \ {T1 , . . . , Tn } that specialize to cohomologically rigid, tame,
irreducible local systems on A1k \ {α1 , . . . , αn } for k algebraically closed.
16
Motivic Nature
Now we look more carefully for a geometric description of these local systems. Our goal will be to
show that these universal rigid local systems have the following form: for any such F, there exists
a smooth family π : X → A1SN,n,` \ {T1 , . . . , Tn } with an action by a finite group Γ, an e ∈ Q` [Γ]
r
s
idempotent, and an r, such that F = GrW
r (eR π! Q` ), and moreover for s 6= r, eR π! Q` = 0.
70
There are various notions of “motivic” we could consider. A very weak notion would be to give
f : Y → A1R \ D smooth such that F ⊆ Rk f! Q` . (In this case, there’s no geometric way to cut F
out.) The above notion is better, but still bad because GrW
r is sheaf-theoretic (`-adic Galois) rather
than geometric. A much better notion would be to have F = eRr π! Q` , but we don’t know this.
Theorem 16.1. Suppose we have two smooth families π : X → A1R \ D and π 0 : X 0 → A1R \ D0 . Let
F and F 0 be lisse on A1R \ D and A1R \ D0 , of the following form: there exist finite groups Γ π and
0
Γ0 π 0 , and idempotents e ∈ Q` [Γ] and e0 ∈ Q` [Γ0 ], such that F = eRk π! Q` and F 0 = e0 Rk π!0 Q` ,
0
and for m 6= k and m0 6= k 0 , eRm π! Q` = 0 and e0 Rm π!0 Q` = 0. Further assume that either F or F 0
is geometrically irreducible and nonconstant.
Form the family
π
e : Xe = (X ×A1 \D A(2)) ×A(2) (X 0 ×A1 \D0 A(2)) → A(2)
(16.1)
0
Then F ?naive F 0 ∼
π )! Q` , and for i 6= m+m0 +1, (e×e0 )Ri (pr2 ◦e
π )! Q` = 0.
= (e×e0 )Rm+m +1 (pr2 ◦e
0
0
That is, F ?naive F has the same form as F, F . Also
0 ∼
0
W
W
GrW
m+m0 +1 (F ?naive F ) = Grm (F) ?mid Grm0 (F ).
(16.2)
Corollary 16.2. Any universal rigid local system F on A1SN,n,` \ {T1 , . . . , Tn } has the form in
the theorem: there exists π : X → A1 \ {T1 , . . . , Tn } smooth with a finite group Γ acting, and e
r
s
idempotent, such that F = GrW
r (eR π! Q` ) and eR π! Q` = 0 for r 6= s.
Remark. Say X/Fq is smooth. Weil I shows that if X is projective, then the action f rq H i (XFq , Q` )
is independent of `. But in general, for X not proper, this is entirely open for H i (X, Q` ) and
Hci (X, Q` ). However, we do know this for Euler characteristics.
Corollary 16.3 (of vanishing in degree 6= r). Independence of ` for F.
Proof of theorem. Form π
e. Künneth implies
Rn π
e! Q` =
M
(Ri π! Q` ⊗ Rj π!0 Q` ).
(16.3)
i+j=n
(We’ll drop the notation for pulling back to A(2).) We find
i
0
eR π! Q` ⊗ e
Rj π!0 Q`
(
0
(e × e0 )Rm+m π
e! Q`
=
0
(i, j) = (m, m0 )
otherwise
(16.4)
Now apply Rpr2! . pr2 is affine, so the Leray spectral sequence
Rp pr2! ◦ Rq π
e! Q` =⇒ Rp+q (pr2 ◦ π
e! )Q`
(16.5)
degenerates at E2 , because E2p,q = 0 unless p = 1, 2. Thus our filtration is
0 ⊆ R2 pr2! (Rn−2 π
e! Q` ) ⊆ Rn (pr2 ◦ π
e)! Q`
with last graded piece R1 pr2! ◦ Rn−1 π
e! Q` . Now applying e × e0 , then
71
(16.6)
e ! Q` ) = R2 pr2! (pr1∗ F ⊗ d∗ F 0 )
R2 pr2! ((e × e0 )Rn−2 pi
(16.7)
if n − 2 = m + m0 (otherwise it vanishes), which is zero by the result last time (using that one of
F or F 0 is geometrically irreducible and nonconstant). The graded part R1 pr2! ((e × e0 )Rn−1 π
e! Q` )
0
vanishes unless n − 1 = m + m , in which case it equals
R1 pr2! (pr1∗ F ⊗ d∗ F 0 ) = F ?naive F 0 .
(16.8)
Now an argument using this similar to last time gives
0
W
W
0
GrW
m+m0 +1 (F ?naive F ) = Grm (F) ?mid Grm0 (F ).
(16.9)
Remark. We’re stuck with having GrW
• , which is geometrically unsatisfying.
We can do better fiber-by-fiber. Let F be a number field, or finitely generated over Q. Then for
any specialization s : Spec F → A1SN,n,` \ {T1 , . . . , Tn }, the π1 (A1SN,n,` \ {T1 , . . . , Tn })-representation
ρF on F pulls back to a Gal(F /F )-representation ρF ,s on s∗ F.
Proposition 16.4. ρF ,s is isomorphic to the `-adic realization of a motivated motive M (F, s) in
MF,Q(ζN ) (for Q(ζN ) the coefficient field, with a fixed embedding into Q` ). Moreover, M (F, s) gives
rise to compatible systems of `-adic representations.
r
Proof. By the previous theorem, ρF ,s ∼
= Grw
r (eHc (XF , Q` )) for some X = Xs /F smooth. That
r
there exists a motivated motive with `-adic realization GrW
r (eHc (XF , Q` )) follows from the explicit
W
geometric description of this Grr , namely
M
r
r
r
∼
,
Q
))
,
Q
)
→
H
(D
,
Q
)
GrW
(H
(X
ker
H
(X
=
`
`
`
r
c
F
F
i,F
where X ,→ X is a smooth normal crossings compactification with complement
the dual of the old Leray spectral sequence argument to compute
H a (X, Rb j∗ Q` ) =⇒ H a+b (X).
(16.10)
S
Di . This is
(16.11)
To incorporate the idempotent e, construct instead a Γ-equivariant compactification. This is
possible: a Γ-equivariant resolution of singularities is known (Abramovich-Wang).
The motto here is that “fibers of rigid local systems are motivated”.
Now we ask about Galois groups. Return to the universal G2 -rigid local system with local
monodromies U (7), −1⊕4 ⊕1⊕3 , U (3)⊕U (2)⊕U (2). A few kinds of “Galois groups” can be considered:
• Take specializations at complex points and ask about Mumford-Tate groups.
• Take specializations at number field points and ask about the Zariski closure of the image of
the Galois representation.
72
• Take specializations at any kind of point of characteristic zero and ask about the motivic
Galois group in the sense of motivated motives.
In all three cases, for “many” specializations, the relevant “Galois group” is G2 . Concretely,
T1 7→ 0, T2 7→ 1, and Z[ N1` , ζN ] ,→ Q (since N = 2 in our setting) gives a local system F on
(A1 \ {0, 1})Q . That is, we have a representation π1 ((A1 \ {0, 1})Q ) → G2 (Q` ) whose geometric
monodromy group (where Q is replaced by Q) is G2 .
Claim. For many s : Spec Q → A1 \ {0, 1}, restriction to
s
Gal(Q/Q)
π1 ((A1 \ {0, 1})Q )
(16.12)
G2 (Q` )
still has monodromy group G2 .
Proofs of specific results follows from specialization results discussed before spring break.
For example, for a “family of motives”, we get nice variational results where generically the
motivic Galois groups is some particular group containing the image of π1geom (here G2 ) as a finite
index subgroup. (This is from André.)
The catch is that a family over A1 \ {0, 1} came from a smooth projective X → A1 \ {0, 1}. In
our setting, X is not projective. There are two ways to get around this:
• (hard) find such a projective X (Dettweiler-Reiter).
• (easy) take a smooth compactification of our smooth family Y → A1 \ {0, 1} over the generic
point of A1 \ {0, 1}, and spread out. Then André applies, possibly replacing A1 \ {0, 1} by
A1 \ {finite number of points}.
Remark.
G2 .
1. The specialization theorem of André actually implies the motivic Galois groups are
2. For the Hodge-theoretic aspect, we actually have a pure variation of Hodge structure over
r
(A1 \ {0, 1})/C from the description F = GrW
r (eR π! Q` ) (the `-adic picture). Using the same
argument of compactifying over the generic point, we get a Betti description over A1 \ {finite}
as
!
e ker Rr π ∗ Q →
M
Rr τi∗ Q
(16.13)
i
where we have
X
S
X
π
π
A1 \ {finite}
73
Di
(16.14)
S
τi
This is visibly a pure weight r variation of Hodge structure. We conclude that over a countable
union of proper analytic subvarieties, the Mumford-Tate group of the fiber of this variation of
Hodge structures is G2 .
A subtlety in the Galois story: (see appendix to Dettweiler-Reiter) For ρF : π1 (A1 \ {0, 1}) →
GL7 (Q` ), is the monodromy group G2 ? By previous arguments, we can drop GL7 to O7 , and then
again to G2 × {±1}. Trivializing the determinant relies on a subtle choice of equations.
17
Alternate Methods
Theorem 17.1 (Z. Yun). Let G be a split, simple, simply connected group of type A1 , D2n , G2 ,
E7 , or E8 . (In other words, the Weyl group of G contains −1 and G is oddly-laced.) Let G∨ be the
dual group (an adjoint group).
(1) There exists a local system ρ : π1 (P1Z[
1
]
2N `
\ {0, 1, ∞}) → G∨ (Q` ) such that for all geometric
points x : Spec k → Spec Z[ 2N1 ` ], the restriction ρx : π1 (P1k \ {0, 1, ∞}) → G∨ (Q` ) has Zariski
dense image except in type D2n . (In this case, we get SO4n−1 .)
(2) For all number fields F and all specializations x : Spec F → P1Z[
ρx : ΓF = Gal(F /F ) → G∨ (Q` ) is “almost” motivated.
1
]
2N `
\ {0, 1, ∞}, the restriction
Precisely, let G∨ → GL(Vqm ) be the quasi-minuscule representation of G∨ . Then ρqm
x : ΓF →
qm
∨
G (Q` ) is such that [ρx |ΓF (i) ⊗ Q` (i)] is isomorphic to the `-adic realization of an object of
MF (i),Q(i) , where M indicates motivated motives.
We only need to adjoin i in types A1 , E7 , and D4n+2 .
Here is the strategy of proof: roughly, a primitive form of geometric Langlands would say that if
k is algebraically closed and X/k is a smooth projective curve, then an “irreducible” G∨ -local system
should arise as the eigen-local system of a Hecke eigensheaf on BunG (X), the moduli of G-bundles
on X.
We have a universal correspondence diagram
Hk
−
→
h
←
−
h
BunG × X
(17.1)
BunG × X
where Hk is the Hecke stack consisting of tuples (P, P 0 , x, ι) where P, P 0 ∈ BunG , x ∈ X, and
∼
ι : P|X\{x} −
→ P 0 |X\{x} .
As source of Hecke operators comes from the geometric Satake category SatG = PervL+ G . Here
LG is the loop group, a functor from k-algebras to groups, mapping R to G(R((t))). L+ G consists
of the positive loops, mapping R to G(R[[t]]). (LG/L+ G = GrG , the affine Grassmannian.) Recall
the Cartan decomposition
G(k((t))) =
a
G(k[[t]])tλ G(k[[t]])
λ∈X∗ (T )+
74
(17.2)
the disjoint union being over dominant cocharacters. This suggests a relation between irreducible
objects of SatG and X∗ (T )+ (which is in bijection with Irr(G∨ )).
Theorem 17.2 (Mirkovic-Vilonen, Ginzburg). For k algebraically closed, there is an equivalence of
∼
Tannakian categories SatG −
→ Rep(G∨ ). Explicitly, to λ ∈ X∗ (T )+ , consider
jλ : GrG,λ ,→ GrG,λ = GrG,≤λ =
[
GrG,µ
(17.3)
µ≤λ
and let ICλ = jλ,!∗ Q` [h2p, λi]. These ICλ ’s are the simple objects of SatG .
We want a version of ICλ on Hk. Then the Hecke operators for every K ∈ SatG are given by
TK
Db (BunG × X)
Db (BunG × X)
→
− ←
−
h ! ( h ∗ F ⊗ KHk ).
F
(17.4)
As for the meaning of KHk : there is a map
L+ G\LG/L+ G
ev : Hk →
AutO
(17.5)
where AutO is the group scheme of continuous automorphisms of k[[t]]. Then KHk = ev∗ K.
A sheaf F ∈ Db (Bun) is a Hecke sheaf, roughly, if there exists a ⊗-functor E : SatG → LS(X)
and “compatible” isomorphisms
∼
αK : TK (F) −
→ E(K) F.
(17.6)
A Hecke sheaf then gives rise to a map π1 (X) → G∨ .
In our setting, take X = P1 and X 0 = P1 \ {0, 1, ∞}. We also no longer require k to be
algebraically closed. We must replace BunG by some moduli of G-bundles with level structures at
the punctures 0, 1, and ∞.
For example, in the A1 case, G = SL2 . The sorts of level structures we care about are Borel
reductions; for instance BunSL2 (B-reduction at 1) would consist of an SL2 -bundle P on XR and a
∼
B-reduction α : PB ×B G −
→ P. We can also interpret this as BunSL2 (full level structure at 1)/I1 ,
for I1 the Iwahori subgroup.
For our moduli space, we have to define certain parahoric level structures. Let Let P 1 ρ∨ ⊆ LG
2
be the parahoric corresponding to the facet containing 21 ρ∨ ∈ X∗ (T )R ∼
= A(T ), where 0 ↔ L+ G. If
G = SL2 , we obtain the Iwahori: X∗ (T ) = Zλ implies 21 ρ∨ = 14 λ.
Let P0 be the subgroup of L0 G in the conjugacy class of P 1 ρ∨ containing I0 , the Iwahori corre2
op
sponding to B. For P∞ , we do the same, except we use the Iwahori I∞
, the Iwahori corresponding
to B op . At 1, take I1 ⊆ L+
G
(for
B).
1
We have to modify this a bit at 0. Function-theoretically, instead of taking automorphic forms
75
.
f : G(F )\G(AF )
×
Y
G(Ox ) × P0 × I1 × P∞ → Q`
(17.7)
x6=0,1,∞
we take forms
.
f : G(F )\G(AF )
Y
×
G(Ox ) × P0+ × I1 × P∞ → Q`
(17.8)
x6=0,1,∞
such that f (gp0 ) = (p0 )f (g) for p0 ∈ P0 , where is the unique quadratic character of P0 (Fq )/P0+ (Fq ) →
∼
±1. For example, on A1 , I0 (Fq )/I0+ (Fq ) −
→ B(Fq )/N (Fq ) = F×
q , which has a unique quadratic character.
Precisely, replace P0 by
(2) P
0
(2) K
0
(17.9)
P0
K0
K0 is the maximal reductive quotient of P0 .
Fact. K0 has a unique connected double cover.
Our moduli of interest is
Bun = BunG ((2) P0 , I1 , P∞ ).
(17.10)
Every object has µker
in its automorphism group.
2
Our eigensheaves will be perverse sheaves on Bun.
Fact. BK0 ⊆ BunG (P0 , P∞ ), by inclusion of the tautological object, is an open substack. The fiber
of this substack under the map
Bun → BunG (P0 , P∞ )
is
(2) K
(17.11)
0 \f lG .
From group theory, P 1 ρ∨ K, the maximal reductive quotient, is actually isomorphic over k
to Gρ
∨ (−1)
2
, of dimension 21 |ΦG | = dim f lG .
Fact. K acts on f lG with finitely many orbits.
Hence K0 f lG has a unique open orbit U ⊆ f lG , and choosing u0 ∈ U (k), we get
(finite group)\ Spec k ∼
= (2) K0 \U ⊆ (2) K0 \f lG ⊆ Bun
(17.12)
so Bun contains an open of the form (finite group)\ Spec k. This suggests rigidity.
For our fixed u0 ∈ U (k), let B0 ∈ f lG (k) correspond to u0 . Then let A = K0 ∩ B0 = stab(u0 ),
a finite group scheme over k. A has a double cover (2) A. This is our relevant finite group; let
j : (2) A\ Spec k → Bun.
76
Here is the main sheaf theoretic result: since µker
is contained in Aut(every object of Bun), we
2
ker
b
get a µ2 -action on sheaves. Then extract D (Bun)odd , the part where µker
acts by its nontrivial
2
character.
Theorem 17.3. The map
∼
j ∗ : Db (Bun)odd −
→ Db ((2) K0 \U )odd
(17.13)
is an equivalence with quasi-inverse j! = j∗ .
It turns out that for any odd central character χ of (2) A(k), there exists a corresponding irreducible representation of (2) A(k). This sets up a bijection between odd irreducible representations
of (2) A(k) and odd central characters, and each of these descends to a representation of (2) A(k)oΓk .
That is, we have a passage from odd χ to LS(2) A (Spec k)odd , which is bijective over k. To our u0 ,
we associate Fχ ∈ LS((2) K0 \U ). Then j! Fχ ∈ Db (Bun) is the desired Hecke eigensheaf.
Roughly, show that TK (j! Fχ ) ∈ Db (X 0 × Bun)odd is perverse up to shift. Then use the explicit
description of Db (Bun)odd to deduce the eigen-property.
Remark. SatG for k a number field can be defined as follows: G is simply connected, and we want to
take direct sums over sheaves ICλ = jλ,!∗ Q` [h2p, λi]. But ICλ ? ICµ may not get a sum of IC’s. We
can get around this by normalizing to ICλ = jλ,!∗ Q` [h2p, λi](hρ, λi). (hρ, λi ∈ Z since G is simply
connected.)
U
The source of motives uses the explicit description of ICλ . We get E(K)|x 99K IHc (GrG,≤λ,x
).
77