Tannakian Categories attached to abelian Varieties
Tannakian Categories attached to abelian
Varieties
Rainer Weissauer
August 22, 2021
Let k be an algebraically closed field k, where k is either the algebraic closure
of a finite field or a field of characteristic zero. Let l be a prime different from the
characteristic of k.
Notations. For a variety X over k let Dbc(X,Ql) denote the triangulated cate-
gory of complexes of etale Ql-sheaves on X in the sense of [5]. For a complex
K ∈ Dbc(X,Ql) let D(K) denote its Verdier dual, and H
ν(K) denote its etale
cohomology Ql-sheaves with respect to the standard t-structure. The abelian
subcategory Perv(X) of middle perverse sheaves is the full subcategory of all
K ∈ Dbc(X,Ql), for which K and its Verdier dual D(K) are contained in the full
subcategory pD≤0(X) of semi-perverse sheaves, where L ∈ Dbc(X,Ql) is semi-
perverse if and only if dim(Sν) ≤ ν holds for all integers ν ∈ Z, where Sν denotes
the support of the cohomology sheaf H−ν(L) of L.
If k is the algebraic closure of a finite field κ, then a complex K of etale Ql-
Weil sheaves is mixed of weight ≤ w, if all its cohomology sheaves Hν(K) are
mixed etale Ql-sheaves with upper weights w(H
ν(K)) − ν ≤ w for all integers ν.
It is called pure of weight w, if K and its Verdier dual D(K) are mixed of weight
≤ w. Concerning base fields of characteristic zero, we assume mixed sheaves to
be sheaves of geometric origin in the sense of the last chapter of [1], so we still
dispose over the notion of the weight filtration and purity and Gabber’s decompo-
sition theorem in this case. In this sense let Pervm(X) denote the abelian category
of mixed perverse sheaves on X. The full subcategory P (X) of Pervm(X) of pure
perverse sheaves is a semisimple abelian category.
http://arxiv.org/abs/0704.1238v2
Abelian varieties. Let X be an abelian variety X of dimension g over an alge-
braically closed field k. The addition law of the abelian variety a : X × X → X
defines the convolution product K ∗ L ∈ Dbc(X,Ql) of two complexes K and L in
Dbc(X,Ql) by the direct image
K ∗ L = Ra∗(K ⊠ L) .
For the skyscraper sheaf δ0 concentrated at the zero element 0 notice K ∗ δ0 = K.
Translation-invariant sheaf complexes. More generally K ∗ δx = T ∗−x(K), where
x is a closed k-valued point in X, δx the skyscraper sheaf with support in {x} and
where Tx(y) = y+x denotes the translation Tx : X → X by x. In fact T ∗y (K ∗L) ∼=
T ∗y (K) ∗ L
∼= K ∗ T ∗y (L) holds for all y ∈ X(k). For K ∈ D
c(X,Ql) let Aut(K)
be the abstract group of all closed k-valued points x of X, for which T ∗x (K) ∼= K
holds. A complex K is called translation-invariant, provided Aut(K) = X(k). If
f : X → Y is a surjective homomorphism between abelian varieties, then the di-
rect image Rf∗(K) of a translation-invariant complex is translation-invariant. As a
consequence of the formulas above, the convolution of an arbitrary K ∈ Dbc(X,Ql)
with a translation-invariant complex on X is a translation-invariant complex. A
translation-invariant perverse sheaf K on X is of the form K = E[g], for an or-
dinary etale translation-invariant Ql-sheaf E. For a translation-invariant complex
K ∈ Dbc(X,Ql) the irreducible constituents of the perverse cohomology sheaves
pHν(K) are translation-invariant.
Multipliers. The subcategory T (X) of Perv(X) of all perverse sheaves, whose ir-
reducible perverse constituents are translation-invariant, is a Serre subcategory of
the abelian category Perv(X). Let denote Perv(X) its abelian quotient category
and P (X) the image of P (X), which is a full subcategory of semisimple objects.
The full subcategory of Dbc(X,Ql) of all K, for which
pHν(K) ∈ T (X), is a thick
subcategory of the triangulated category Dbc(X,Ql). Let
c(X,Ql)
be the corresponding triangulated quotient category, which contains Perv(X).
Then the convolution product
∗ : D
c(X,Ql)×D
c(X,Ql) → D
c(X,Ql)
still is well defined, by reasons indicated above.
Definition. A perverse sheaf K on X is called a multiplier, if the convolution
induced by K
∗K : Dbc(X,Ql) → D
c(X,Ql)
preserves the abelian subcategory Perv(X).
Obvious from this definition are the following properties of multipliers: If K
and L are multipliers, so are the product K ∗ L and the direct sum K ⊕ L. Direct
summands of multipliers are multipliers. If K is a multiplier, then the Verdier dual
D(K) is a multiplier and also the dual
K∨ = (−idX)
∗(D(K)) .
Examples: 1) Skyscraper sheaves are multipliers 2) If i : C →֒ X is a projective
curve, which generates the abelian variety X, and E is an etale Ql-sheaf on C
with finite monodromy, then the intersection cohomology sheaf attached to (C,E)
is a multiplier. 3) If : Y →֒ X is a smooth ample divisor, then the intersection
cohomology sheaf of Y is a multiplier.
The proofs. 1) is obvious. For 2) we gave in [7] a proof by reduction mod p
using the Cebotarev density theorem and counting of points. Concerning 3) the
morphism j : U = X \Y →֒ X is affine for ample divisors Y . Hence λU = Rj!Ql[g]
and λY = i∗Ql,Y [g − 1] are perverse sheaves, which coincide in Perv(X). The
morphism π = a◦(j×idX ) is affine. Indeed W = π−1(V ) is affine for affine subsets
V of X, W being isomorphic under the isomorphism (u, v) 7→ (u, u + v) of X2 to
the affine product U × V . By the affine vanishing theorem of Artin: For perverse
sheaves L ∈ Perv(X) we get λU ⊠ L ∈ Perv(X2) and pHν(Rπ!(λU ⊠ L)) = 0 for
all ν < 0. The distinguished triangle
Ra∗(λY ⊠ L), Rπ!(λU ⊠ L), Ra∗(δX ⊠ L)
for δX = Ql,X [g] and the corresponding long exact perverse cohomology sequence
gives isomorphisms pHν−1(δX ∗ L) ∼= pHν(λY ∗ L) for the integers ν < 0. Since
Ra∗(δX ⊠ L) = δX ∗ L is a direct sum of translates of constant perverse sheaves
δX , we conclude pHν(λY ∗ L) for ν < 0 to be zero in Perv(X). For smooth Y the
intersection cohomology sheaf is λY = i∗Ql,Y [g − 1], and it is self dual. Hence
by Verdier duality i∗Ql,Y [g − 1] ∗ L has image in Perv(X). Thus i∗Ql,Y [g − 1] is a
multiplier. �
Let M(X) ⊆ P (X) denote the full category of semisimple multipliers. Let
M(X) denote its image in the quotient category P (X) of P (X). Then, by the
definition of multipliers, the convolution product preserves M(X)
∗ : M(X)×M(X) → M(X) .
Theorem. With respect to this convolution product the category M(X) is a
semisimple super-Tannakian Ql-linear tensor category, hence as a tensor cate-
gory M(X) is equivalent to the category of representations Rep(G, ε) of a projec-
tive limit
G = G(X)
of supergroups.
Outline of proof. The convolution product obviously satisfies the usual commuta-
tivity and associativity constraints compatible with unit objects. See [7] 2.1. By
[7], corollary 3 furthermore one has functorial isomorphisms
(K,L) ∼= Γ{0}(X,H
0(K ∗ L∨)∗) ,
where H0 denotes the degree zero cohomology sheaf and Γ{0}(X,−) sections with
support in the neutral element. Let L = K be simple and nonzero. Then the left
side becomes End
M(X)(K)
∼= Ql. On the other hand K ∗ L∨ is a direct sum of a
perverse sheaf P and translates of translation-invariant perverse sheaves. Hence
H0(K ∗ L∨)∨) is the direct sum of a skyscraper sheaf S and translation-invariant
etale sheaves. Therefore Γ{0}(X,H
0(K ∗ L∨)∨) = Γ{0}(X,S). By a comparison
of both sides therefore S = δ0. Notice δ0 is the unit element 1 of the convolution
product. Using the formula above we not only get
(K,L) ∼= HomM(X)(K ∗ L
∨, 1) ,
but also find a nontrivial morphism
evK : K ∗K
∨ → 1 .
By semisimplicity δ0 is a direct summand of the complex K ∗ K∨. In particular
the Künneth formula implies, that the etale cohomology groups do not all vanish
identically
H•(X,K) 6= 0 .
Therefore the arguments of [7] 2.6 show, that the simple perverse sheaf K is du-
alizable. Hence M(X) is a rigid Ql-linear tensor category. Let T be a finitely
⊗-generated tensor subcategory with generator say A. To show T is super-
Tannakian, by [4] it is enough to show for all n
lenghtT (A
∗n) ≤ Nn ,
where N is a suitable constant. For any B ∈ M(X) let B, by abuse of no-
tation, also denote the perverse semisimple representative in Perv(X) without
translation invariant summand. Put h(B, t) =
ν dimQl(H
ν(X,B))tν . Then
lenghtT (B) ≤ h(B, 1), since every summand of B is a multiplier and there-
fore has nonvanishing cohomology. For B = A∗n the Künneth formula gives
h(B, 1) = h(A, 1)n. Therefore the estimate above holds for N = h(A, 1). This
completes the outline for the proof of the theorem. �
Principally polarized abelian varieties. Suppose Y is a divisor in X defining a
principal polarization. Suppose the intersection cohomology sheaf δY of Y is a
multiplier. Then a suitable translate of Y is symmetric, and again a multiplier. So
we may assume Y = −Y is symmetric. Let M(X,Y ) denote the super-Tannakian
subcategory of M(X) generated by δY . The corresponding super-group G(X,Y )
attached to M (X,Y ) acts on the super-space W = ω(δY ) defined by the underlying
super-fiber functor ω of M(X). By assumption δY is self dual in the sense, that
there exists an isomorphism ϕ : δ∨
∼= δY . Obviously ϕ∨ = ±ϕ. This defines
a nondegenerate pairing on W , and the action of G(X,Y ) on W respects this
pairing.
Curves. If X is the Jacobian of smooth projective curve C of genus g over k, X car-
ries a natural principal polarization Y = Wg−1. If we replace this divisor by a sym-
metric translate, then Y is a multiplier. The corresponding group G(X,Y ) is the
semisimple algebraic group G = Sp(2g−2,Ql)/µg−1[2] or G = Sl(2g−2,Ql)/µg−1
depending on whether the curve C is hyperelliptic or not. The representation W
of G(X,Y ) defined by δY as above is the unique irreducible Ql-representation of
G(X,Y ) of highest weight, which occurs in the (g − 1)-th exterior power of the
(2g − 2)-dimensional standard representation of G. See [7], section 7.6.
Conjecture. One could expect, that a principal polarized abelian variety (X,Y )
of dimension g is isomorphic to a Jacobian variety (Jac(C),Wg−1) of a smooth
projective curve C (up to translates of the divisor Y in X as explained above) if
and only if Y is a multiplier with corresponding super-Tannakian group G(X,Y )
equal to one of the two groups
Sp(2g − 2,Ql)/µg−1[2] or Sl(2g − 2,Ql)/µg−1 .
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http://arxiv.org/abs/math/0610460
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