On the Tannaka group attached to the Theta divisor of a generic principally polarized abelian variety

Abstract

To any closed subvariety Y of a complex abelian variety one can attach a reductive algebraic group G which is determined by the decomposition of the convolution powers of Y via a certain Tannakian formalism. For a theta divisor Y on a principally polarized abelian variety, this group G provides a new invariant that naturally endows the moduli space ${{\mathcal {A}}}_g$ of principally polarized abelian varieties of dimension g with a finite constructible stratification. We determine G for a generic principally polarized abelian variety, and for $g=4$ we show that the stratification detects the locus of Jacobian varieties inside the moduli space of abelian varieties.

Appendix: Constructibility

In this appendix we study how the Tannaka groups from [27] vary in families of perverse sheaves. We work in an algebraic setting, so as above we put $\Lambda = \overline{\mathbb {Q}}_l$ for some prime l and write ${\mathbf {P}}(-)=\mathrm {Perv}(-, \Lambda )$ and ${\mathbf {D}}(-)={{D^{b}_{c} }}(-, \Lambda )$.

Let S be an algebraic variety over an algebraically closed field k of characteristic zero, and let ${\mathcal {X}}\rightarrow S$ be an abelian scheme over S. If ${\mathcal {Y}}\hookrightarrow {\mathcal {X}}$ is a closed subvariety that maps surjectively onto S, we can consider for any geometric point $\overline{s}$ in S the perverse intersection cohomology sheaf $\delta _{\, Y_{\overline{s}}} \in {\mathbf {P}}(X_{\overline{s}})$ on the fibre $Y_{\overline{s}} \hookrightarrow X_{\overline{s}}$. The following example illustrates that the corresponding Tannaka groups in general do not depend on $\overline{s}$ in a constructible way.

Example 7.1

If $S=E$ is an elliptic curve and ${\mathcal {X}}=E\times S \rightarrow S$ is the constant family, then for the diagonal ${\mathcal {Y}}= \{ (e,e) \mid e\in E\}$ we have

$$\begin{aligned} G(\delta _{\, Y_{\overline{s}}}) \; \cong \; {\left\{ \begin{array}{ll} {\mathbb {Z}}/n{\mathbb {Z}}&{} \hbox { if } \overline{s}\hbox { is a torsion point in } E \hbox { of precise order } n, \\ {\mathbb {G}}_m &{} \hbox { if } \overline{s}\hbox { is a point of infinite order in } E. \end{array}\right. } \end{aligned}$$

In general we only have the following semicontinuity property.

Lemma 7.2

Let ${\mathcal {Y}}\hookrightarrow {\mathcal {X}}$ be a closed subvariety which is smooth over S. Let $\eta \in S$ be a scheme-theoretic point, $s \in \overline{\{ \eta \}}$ a point in its closure, and choose geometric points $\bar{\eta }$ and $\overline{s}$ above them. Then we have an embedding

$$\begin{aligned} G(\delta _{\, Y_{\overline{s}}}) \hookrightarrow G(\delta _{\, Y_{ \bar{\eta }}}). \end{aligned}$$

Proof

After base change to the reduced closed subscheme $\overline{\{ \eta \}} \hookrightarrow S$ we can assume that $\eta$ is the generic point of S. One then easily reduces our claim to the local situation of [27, lemma 14.1], and for the nearby cycles we have $\Psi (\delta _{Y_{ \bar{\eta }}}) = \delta _{Y_{\overline{s}}}$ because by assumption the morphism ${\mathcal {Y}}\rightarrow S$ is smooth. $\square$

To get a constructibility statement for the stratifications defined by the Tannaka groups, we need to impose some finiteness conditions. To simplify the notation, let us temporarily assume that ${\mathcal {X}}=X$ is an abelian variety over $S= Spec (k)$. For every $P\in {\mathbf {P}}(X)$ we have a fibre functor

$$\begin{aligned} \omega : \langle P \rangle \; \mathop {\longrightarrow }\limits ^{\sim } \; { Rep }_\Lambda (G(P)). \end{aligned}$$

Like any character of the Tannaka group, the determinant character $\det (\omega (P))$ corresponds by [27, prop. 10.1] to a skyscraper sheaf $\delta _{x} \in \langle P \rangle$ supported on some point $x \in X(k)$. By abuse of notation, in what follows we write $\det (P)^n={\mathbf {1}}$ to indicate that the determinant character has order dividing n, or equivalently that the above point x is an n-torsion point. In many applications the perverse sheaf P will be isomorphic to its adjoint dual $P^\vee$ and then $\det (P)^2 = {\mathbf {1}}$; by Remark 1.1(c) this in particular holds if $P=\delta _Y$ is the perverse intersection cohomology sheaf supported on a symmetric closed subvariety $Y\hookrightarrow X$.

Lemma 7.3

For any fixed $d, n\in {\mathbb {N}}$, the general linear group $Gl _d(\Lambda )$ admits only finitely many conjugacy classes of subgroups isomorphic to Tannaka groups G(P) of simple perverse sheaves $P\in {\mathbf {P}}(X)$ with $\chi (P)=d$ and $\det (P)^n={\mathbf {1}}$.

Proof

Let $G=G(P)$ be the Tannaka group of a simple perverse sheaf P as in the lemma, and denote by $H=G^0$ the connected component of this Tannaka group. By [35] there exists a finite subgroup $K\subset X$ of torsion points on X such that we have

$$\begin{aligned} H\;=\; G\bigl (Rf_*(P) \bigr ) \quad \hbox {for the isogeny} \quad f: \; X \; \rightarrow \; X/K, \end{aligned}$$

and the group of connected components $\pi = G/H$ is a finite abelian group whose characters correspond to the rank one skyscraper sheaves supported in the points of $K({\mathbb {C}})$. The restriction and induction functors for the subgroup $H\hookrightarrow G$ are given by the direct and inverse image:

and looking at the restriction of the induction of a character $\chi : H \rightarrow \Lambda ^*$ one sees that the adjoint action of G fixes any such character. In other words, the adjoint action of G induces the trivial action on the abelianization $H^{ab} = H/[H, H]$. So we may apply the following general fact:

Let V be a finite-dimensional vector space over $\Lambda$. Then for any fixed $n\in {\mathbb {N}}$, there exist only finitely many conjugacy classes of subgroups $G\hookrightarrow Gl (V)$ with the property that

• (1) the restriction $V|_G$ is irreducible,

• (2) the quotient $\pi = G/H$ by $H=G^0$ is a finite abelian group,

• (3) the adjoint action of $\pi$ on $H^{ab} = H/[H, H]$ is trivial,

• (4) the determinant $\det (V|_G)$ is a character of order at most n.

For the proof of this general fact one may fix the subgroup $H\hookrightarrow Gl (V)$ since the group $Gl (V)$ contains only finitely many connected reductive subgroups up to conjugation. Using Mackey theory and the classification of non-connected reductive groups, one then shows that there are only finitely many subgroups $G\hookrightarrow Gl (V)$ with given connected component $G^0 = H$ such that (1)–(4) hold; for details we refer to [25, appendix B]. $\square$

Let us now return to an abelian scheme $p: {\mathcal {X}}\rightarrow S$ whose base scheme is any variety S over k. For $K\in {\mathbf {D}}(X)$ and geometric points $\overline{s}$ of S we write $K_{\overline{s}} = i_{\overline{s}}^* (K)$ for the pull-back to the geometric fibre $i_{\overline{s}}: X_{\overline{s}} \rightarrow {\mathcal {X}}$.

Proposition 7.4

Let $n\in {\mathbb {N}}$ and $P\in {\mathbf {D}}({\mathcal {X}})$ be such that for any geometric point $\overline{s}$ the pull-back $P_{\overline{s}}$ is a simple perverse sheaf with $\det (P_{\overline{s}})^n = {\mathbf {1}}$. Then there are reductive algebraic groups $G_1, \dots , G_m$ and an algebraic stratification into locally closed subsets

$$\begin{aligned} S \;=\; \bigsqcup _{i=0}^m \; S_i \;\;\; \hbox { such that} \;\;\; G(P_{\overline{s}}) \;\cong \; G_i \;\;\; \hbox { for all geometric points } \overline{s}\hbox { in } S_i. \end{aligned}$$

Proof

If V is a finite-dimensional vector space over $\Lambda$, then every reductive algebraic subgroup of $Gl _\Lambda (V)$ is determined uniquely by its invariants in the tensor powers $W^{\otimes r}$ of the representation $W=V\oplus V^\vee$ [12, prop. 3.1(c)]. Furthermore, if we only want to distinguish between finitely many given reductive subgroups up to conjugacy, then it suffices to consider only finitely many exponents $r \in {\mathbb {N}}$.

In our case it follows via Lemma 7.3 that the group $G(P_{\overline{s}})$ is determined by the collection of all direct summands $\delta _0$ inside the convolution powers $(P_{\overline{s}} \oplus (P_{\overline{s}})^\vee )^{*r}$ for finitely many r. To show that these direct summands depend on the geometric point $\overline{s}$ in a constructible way, we may replace S by an open dense subset and then proceed by induction on $\dim (S)$. So we may assume that there exists $K \in {\mathbf {D}}({\mathcal {X}})$ such that

$$\begin{aligned} K_{\overline{s}} \;\cong \; P_{\overline{s}} \oplus (P_{\overline{s}})^\vee \quad \hbox {for all geometric points } \overline{s}\hbox { in } S. \end{aligned}$$

Here we use the general fact [23, prop. 1.1.7] that for any morphism $p: {\mathcal {X}}\rightarrow S$ with smooth target S and any constructible complex $P\in {\mathbf {D}}({\mathcal {X}})$, there is an open dense subset of S such that the formation of the relative Verdier dual ${R\mathcal {H}om}(P, p^{!}\Lambda _S)$ commutes with any base change that factors over this open subset. Now consider the relative convolution powers

$$\begin{aligned} K(r) \;=\; Ra_{r,*}(K\boxtimes \cdots \boxtimes K) \end{aligned}$$

defined by the r-fold addition morphism $a_r: {\mathcal {X}}\times _S \cdots \times _S {\mathcal {X}}\rightarrow {\mathcal {X}}$ of our abelian scheme. These relative convolution powers are constructible sheaf complexes, hence in particular the dimension $\dim _\Lambda { Hom }(\delta _0, K(r)_{\overline{s}})$ is a constructible function of the geometric point $\overline{s}$ in S. The isomorphism

$$\begin{aligned} (K(r))_{\overline{s}} \;\cong \; (P_{\overline{s}} \oplus (P_{\overline{s}})^\vee )^{*r} \end{aligned}$$

and the remarks from the beginning of the proof thus imply that the isomorphism class of the group $G(P_{\overline{s}})$ is a constructible function of the point $\overline{s}$ in S. $\square$