Tannakian Categories attached to abelian Varieties

TL;DR

This paper constructs a semisimple super-Tannakian category from specific perverse sheaves on abelian varieties, including intersection cohomology sheaves of curves and smooth ample divisors.

Contribution

It introduces a new categorical framework linking perverse sheaves on abelian varieties to Tannakian categories, expanding the understanding of their geometric and algebraic structures.

Findings

Construction of a semisimple super-Tannakian category from perverse sheaves

Inclusion of intersection cohomology sheaves of curves and ample divisors

Establishment of categorical properties related to abelian varieties

Abstract

Starting from certain perverse sheaves on an abelian variety, including the intersection cohomology sheaves of curves and smooth ample divisors, we construct a semisimple super-Tannakian category.

Equations

$$K\ast L=R{a}_{\ast }(K⊠L).$$

$${\overline{D}}_c^b(X,{\overline{\mathbb{Q}}}_l)$$

$$\ast :{\overline{D}}_c^b(X,{\overline{\mathbb{Q}}}_l)\times {\overline{D}}_c^b(X,{\overline{\mathbb{Q}}}_l)\to {\overline{D}}_c^b(X,{\overline{\mathbb{Q}}}_l)$$

$$\ast K:\overline{D_c^b}(X,\overline{{\mathbb{Q}}_l})\to \overline{D_c^b}(X,\overline{{\mathbb{Q}}_l})$$

$$K^{\vee }=(-i{d}_X{)}^{\ast }(D(K)).$$

$$\ast :\overline{M}(X)\times \overline{M}(X)\to \overline{M}(X).$$

$$G=G(X)$$

$$Ho{m}_{\overline{M}(X)}(K,L)\cong {\Gamma }_{\{0\}}(X,{\mathcal{H}}^0(K\ast L^{\vee }{)}^{\ast }),$$

$$Ho{m}_{\overline{M}(X)}(K,L)\cong Ho{m}_{\overline{M}(X)}(K\ast L^{\vee },1),$$

$$e{v}_K:K\ast K^{\vee }\to 1.$$

$$H^{\bullet }(X,K)\ne 0.$$

$$lengh{t}_{\mathcal{T}}(A^{\ast n})\le N^n,$$

$$Sp(2g-2,{\overline{\mathbb{Q}}}_l)/{\mu }_{g-1}[2]\text{mbox}orSl(2g-2,{\overline{\mathbb{Q}}}_l)/{\mu }_{g-1}.$$

Full text

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Tannakian Categories attached to abelian Varieties

Rainer Weissauer

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 $D_{c}^{b}(X,\overline{\mathbb{Q}}_{l})$ denote the triangulated category of complexes of etale $\overline{\mathbb{Q}}_{l}$-sheaves on $X$ in the sense of [5]. For a complex $K\in D_{c}^{b}(X,\overline{\mathbb{Q}}_{l})$ let $D(K)$ denote its Verdier dual, and ${\cal H}^{\nu}(K)$ denote its etale cohomology $\overline{\mathbb{Q}}_{l}$-sheaves with respect to the standard $t$-structure. The abelian subcategory $Perv(X)$ of middle perverse sheaves is the full subcategory of all $K\in D_{c}^{b}(X,\overline{\mathbb{Q}}_{l})$, for which $K$ and its Verdier dual $D(K)$ are contained in the full subcategory ${}^{p}\!D^{\leq 0}(X)$ of semi-perverse sheaves, where $L\in D_{c}^{b}(X,\overline{\mathbb{Q}}_{l})$ is semi-perverse if and only if $dim(S_{\nu})\leq\nu$ holds for all integers $\nu\in\mathbb{Z}$, where $S_{\nu}$ denotes the support of the cohomology sheaf ${\cal H}^{-\nu}(L)$ of $L$.

If $k$ is the algebraic closure of a finite field $\kappa$, then a complex $K$ of etale $\overline{\mathbb{Q}}_{l}$-Weil sheaves is mixed of weight $\leq w$, if all its cohomology sheaves ${\cal H}^{\nu}(K)$ are mixed etale $\overline{\mathbb{Q}}_{l}$-sheaves with upper weights $w({\cal H}^{\nu}(K))-\nu\leq w$ for all integers $\nu$. It is called pure of weight $w$, if $K$ and its Verdier dual $D(K)$ are mixed of weight $\leq w$. 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 decomposition theorem in this case. In this sense let $Perv_{m}(X)$ denote the abelian category of mixed perverse sheaves on $X$. The full subcategory $P(X)$ of $Perv_{m}(X)$ of pure perverse sheaves is a semisimple abelian category.

Abelian varieties. Let $X$ be an abelian variety $X$ of dimension $g$ over an algebraically closed field $k$. The addition law of the abelian variety $a:X\times X\to X$ defines the convolution product $K*L\in D_{c}^{b}(X,\overline{\mathbb{Q}}_{l})$ of two complexes $K$ and $L$ in $D_{c}^{b}(X,\overline{\mathbb{Q}}_{l})$ by the direct image

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For the skyscraper sheaf $\delta_{0}$ concentrated at the zero element [math] notice $K*\delta_{0}=K$.

Translation-invariant sheaf complexes. More generally $K*\delta_{x}=T^{*}_{-x}(K)$, where $x$ is a closed $k$-valued point in $X$, $\delta_{x}$ the skyscraper sheaf with support in $\{x\}$ and where $T_{x}(y)=y+x$ denotes the translation $T_{x}:X\to X$ by $x$. In fact $T^{*}_{y}(K*L)\cong T^{*}_{y}(K)*L\cong K*T^{*}_{y}(L)$ holds for all $y\in X(k)$. For $K\in D_{c}^{b}(X,\overline{\mathbb{Q}}_{l})$ let $Aut(K)$ be the abstract group of all closed $k$-valued points $x$ of $X$, for which $T^{*}_{x}(K)\cong K$ holds. A complex $K$ is called translation-invariant, provided $Aut(K)=X(k)$. If $f:X\to Y$ is a surjective homomorphism between abelian varieties, then the direct image $Rf_{*}(K)$ of a translation-invariant complex is translation-invariant. As a consequence of the formulas above, the convolution of an arbitrary $K\in D_{c}^{b}(X,\overline{\mathbb{Q}}_{l})$ 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 ordinary etale translation-invariant $\overline{\mathbb{Q}}_{l}$-sheaf $E$. For a translation-invariant complex $K\in D_{c}^{b}(X,\overline{\mathbb{Q}}_{l})$ the irreducible constituents of the perverse cohomology sheaves ${}^{p}H^{\nu}(K)$ are translation-invariant.

Multipliers. The subcategory $T(X)$ of $Perv(X)$ of all perverse sheaves, whose irreducible perverse constituents are translation-invariant, is a Serre subcategory of the abelian category $Perv(X)$. Let denote $\overline{Perv}(X)$ its abelian quotient category and $\overline{P}(X)$ the image of $P(X)$, which is a full subcategory of semisimple objects. The full subcategory of $D_{c}^{b}(X,\overline{\mathbb{Q}}_{l})$ of all $K$, for which ${}^{p}H^{\nu}(K)\in T(X)$, is a thick subcategory of the triangulated category $D_{c}^{b}(X,\overline{\mathbb{Q}}_{l})$. Let

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be the corresponding triangulated quotient category, which contains $\overline{Perv}(X)$. Then the convolution product

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still is well defined, by reasons indicated above.

Definition. A perverse sheaf $K$ on $X$ is called a multiplier, if the convolution induced by $K$

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preserves the abelian subcategory $\overline{Perv}(X)$.

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 $K\oplus L$. Direct summands of multipliers are multipliers. If $K$ is a multiplier, then the Verdier dual $D(K)$ is a multiplier and also the dual

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Examples: 1) Skyscraper sheaves are multipliers 2) If $i:C\hookrightarrow X$ is a projective curve, which generates the abelian variety $X$, and $E$ is an etale $\overline{\mathbb{Q}}_{l}$-sheaf on $C$ with finite monodromy, then the intersection cohomology sheaf attached to $(C,E)$ is a multiplier. 3) If $:Y\hookrightarrow X$ 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 $j:U=X\setminus Y\hookrightarrow X$ is affine for ample divisors $Y$. Hence $\lambda_{U}=Rj_{!}\overline{\mathbb{Q}}_{l}[g]$ and $\lambda_{Y}=i_{*}\overline{\mathbb{Q}}_{l,Y}[g-1]$ are perverse sheaves, which coincide in $\overline{Perv}(X)$. The morphism $\pi=a\circ(j\times id_{X})$ is affine. Indeed $W=\pi^{-1}(V)$ is affine for affine subsets $V$ of $X$, $W$ being isomorphic under the isomorphism $(u,v)\mapsto(u,u+v)$ of $X^{2}$ to the affine product $U\times V$. By the affine vanishing theorem of Artin: For perverse sheaves $L\in Perv(X)$ we get $\lambda_{U}\boxtimes L\in Perv(X^{2})$ and ${}^{p}H^{\nu}(R\pi_{!}(\lambda_{U}\boxtimes L))=0$ for all $\nu<0$. The distinguished triangle \bigl{(}Ra_{*}(\lambda_{Y}\boxtimes L),R\pi_{!}(\lambda_{U}\boxtimes L),Ra_{*}(\delta_{X}\boxtimes L)\bigr{)} for $\delta_{X}=\overline{\mathbb{Q}}_{l,X}[g]$ and the corresponding long exact perverse cohomology sequence gives isomorphisms ${}^{p}H^{\nu-1}(\delta_{X}*L)\cong{}^{p}H^{\nu}(\lambda_{Y}*L)$ for the integers $\nu<0$. Since $Ra_{*}(\delta_{X}\boxtimes L)=\delta_{X}*L$ is a direct sum of translates of constant perverse sheaves $\delta_{X}$, we conclude ${}^{p}H^{\nu}(\lambda_{Y}*L)$ for $\nu<0$ to be zero in $\overline{Perv}(X)$. For smooth $Y$ the intersection cohomology sheaf is $\lambda_{Y}=i_{*}\overline{\mathbb{Q}}_{l,Y}[g-1]$, and it is self dual. Hence by Verdier duality $i_{*}\overline{\mathbb{Q}}_{l,Y}[g-1]*L$ has image in $\overline{Perv}(X)$. Thus $i_{*}\overline{\mathbb{Q}}_{l,Y}[g-1]$ is a multiplier. $\quad\square$

Let $M(X)\subseteq P(X)$ denote the full category of semisimple multipliers. Let ${\overline{M}}(X)$ denote its image in the quotient category $\overline{P}(X)$ of $P(X)$. Then, by the definition of multipliers, the convolution product preserves ${\overline{M}}(X)$

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Theorem. With respect to this convolution product the category ${\overline{M}}(X)$ is a semisimple super-Tannakian $\overline{\mathbb{Q}}_{l}$-linear tensor category, hence as a tensor category ${\overline{M}}(X)$ is equivalent to the category of representations $Rep(G,\varepsilon)$ of a projective limit

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of supergroups.

Outline of proof. The convolution product obviously satisfies the usual commutativity and associativity constraints compatible with unit objects. See [7] 2.1. By [7], corollary 3 furthermore one has functorial isomorphisms

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where ${\cal H}^{0}$ denotes the degree zero cohomology sheaf and $\Gamma_{\{0\}}(X,-)$ sections with support in the neutral element. Let $L=K$ be simple and nonzero. Then the left side becomes $End_{{\overline{M}}(X)}(K)\cong\overline{\mathbb{Q}_{l}}$. On the other hand $K*L^{\vee}$ is a direct sum of a perverse sheaf $P$ and translates of translation-invariant perverse sheaves. Hence ${\cal H}^{0}(K*L^{\vee})^{\vee})$ is the direct sum of a skyscraper sheaf $S$ and translation-invariant etale sheaves. Therefore $\Gamma_{\{0\}}(X,{\cal H}^{0}(K*L^{\vee})^{\vee})=\Gamma_{\{0\}}(X,S)$. By a comparison of both sides therefore $S=\delta_{0}$. Notice $\delta_{0}$ is the unit element $1$ of the convolution product. Using the formula above we not only get

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but also find a nontrivial morphism

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By semisimplicity $\delta_{0}$ is a direct summand of the complex $K*K^{\vee}$. In particular the Künneth formula implies, that the etale cohomology groups do not all vanish identically

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Therefore the arguments of [7] 2.6 show, that the simple perverse sheaf $K$ is dualizable. Hence ${\overline{M}}(X)$ is a rigid $\overline{\mathbb{Q}_{l}}$-linear tensor category. Let ${\cal T}$ be a finitely $\otimes$-generated tensor subcategory with generator say $A$. To show ${\cal T}$ is super-Tannakian, by [4] it is enough to show for all $n$

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where $N$ is a suitable constant. For any $B\in{\overline{M}}(X)$ let $B$, by abuse of notation, also denote the perverse semisimple representative in $Perv(X)$ without translation invariant summand. Put $h(B,t)=\sum_{\nu}dim_{\overline{\mathbb{Q}_{l}}}(H^{\nu}(X,B))t^{\nu}$. Then $lenght_{\cal T}(B)\leq h(B,1)$, since every summand of $B$ is a multiplier and therefore 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. $\quad\square$

Principally polarized abelian varieties. Suppose $Y$ is a divisor in $X$ defining a principal polarization. Suppose the intersection cohomology sheaf $\delta_{Y}$ 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 ${\overline{M}}(X,Y)$ denote the super-Tannakian subcategory of $\overline{M}(X)$ generated by $\delta_{Y}$. The corresponding super-group $G(X,Y)$ attached to ${\overline{M}}(X,Y)$ acts on the super-space $W=\omega(\delta_{Y})$ defined by the underlying super-fiber functor $\omega$ of $\overline{M}(X)$. By assumption $\delta_{Y}$ is self dual in the sense, that there exists an isomorphism $\varphi:\delta_{Y}^{\vee}\cong\delta_{Y}$. Obviously $\varphi^{\vee}=\pm\varphi$. 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 $k$, $X$ carries a natural principal polarization $Y=W_{g-1}$. If we replace this divisor by a symmetric translate, then $Y$ is a multiplier. The corresponding group $G(X,Y)$ is the semisimple algebraic group $G=Sp(2g-2,\overline{\mathbb{Q}}_{l})/\mu_{g-1}[2]$ or $G=Sl(2g-2,\overline{\mathbb{Q}}_{l})/\mu_{g-1}$ depending on whether the curve $C$ is hyperelliptic or not. The representation $W$ of $G(X,Y)$ defined by $\delta_{Y}$ as above is the unique irreducible $\overline{\mathbb{Q}}_{l}$-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 $(Jac(C),W_{g-1})$ 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

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