Characteristic classes and Hilbert-Poincaré series

Characteristic classes and Hilbert-Poincaré series for perverse sheaves on abelian varieties

Thomas Krämer Mathematisches Institut
Ruprecht-Karls-Universität Heidelberg
Im Neuenheimer Feld 205, D-69120 Heidelberg, Germany

The convolution powers of a perverse sheaf on an abelian variety define an interesting family of branched local systems whose geometry is still poorly understood. We show that the generating series for their generic rank is a rational function of a very simple shape and that a similar result holds for the symmetric convolution powers. We also give formulae for other Schur functors in terms of characteristic classes on the dual abelian variety, and as an example we discuss the case of Prym-Tjurin varieties.

Key words and phrases:
Chern-MacPherson class, perverse sheaf, convolution product, abelian variety, Schur functor, symmetric power, generating series, tensor category.
2010 Mathematics Subject Classification:
Primary 14K99; Secondary 14C17, 18D10, 32S60

1. Introduction

Tannakian categories arise naturally in many areas of algebraic geometry. For instance, Gabber and Loeser have studied convolutions of perverse sheaves on tori via certain Tannakian quotient categories [GaL] which have also been used recently by Katz in his work on the Mellin transform over finite fields [KatzSatoTate]. In what follows we will be concerned with complex abelian varieties, where similar results have been obtained in relation with a generic vanishing theorem for perverse sheaves [KrWVanishing]. The Tannakian formalism produces many new objects from a given one by considering convolution powers and their subquotients such as symmetric powers; the arising perverse sheaves encode nontrivial information on the geometry of moduli spaces and Albanese morphisms [KrCubic] [KrE6] [KrWSchottky] [WeBN] [WeTorelli], but they are hard to approach explicitly. The goal of this note is to study their generic rank. It turns out that the generating series for the symmetric powers is a rational function of a very simple shape reminiscent of the Hilbert-Poincaré series of a graded module.

Let A be a complex abelian variety, and denote by \operatorname{D^{\mathit{b}}_{\mathit{c}}}(A)=\operatorname{D^{\mathit{b}}_{% \mathit{c}}}({\mathbb{C}}_{A}) the bounded derived category of the category of constructible sheaves of complex vector spaces on A. The group law a:A\times A\to A of the abelian variety defines a convolution product

K*L\;=\;Ra_{*}(K\boxtimes L)\quad\textnormal{for}\quad K,L\in\operatorname{D^{% \mathit{b}}_{\mathit{c}}}(A),

and with respect to this convolution product the derived category becomes a rigid symmetric monoidal category. The abelian subcategory \operatorname{P}(A)\subset\operatorname{D^{\mathit{b}}_{\mathit{c}}}(A) of perverse sheaves is not stable under convolution, but one gets a Tannakian quotient category as follows. By the vanishing theorem of [KrWVanishing] [SchnellHolonomic] the Euler characteristic of perverse sheaves is nonnegative:

\chi(P)\;\;=\;\;\sum_{n\in{\mathbb{Z}}}\;(-1)^{n}\dim_{\mathbb{C}}% \operatorname{H}^{n}(A,P)\;\;\geq\;\;0\quad\textnormal{for all}\quad P\in% \operatorname{P}(A).

We say that a complex K\in\operatorname{D^{\mathit{b}}_{\mathit{c}}}(A) is negligible if the Euler characteristic of each of its perverse cohomology sheaves vanishes. The negligible complexes form a thick subcategory \operatorname{N}(A)\subset\operatorname{D^{\mathit{b}}_{\mathit{c}}}(A), and the quotient category \operatorname{P}(A)/(\operatorname{P}(A)\cap\operatorname{N}(A)) is a limit of Tannakian categories with a tensor product induced from the convolution product [KrWVanishing]. If \operatorname{{P_{\hskip-2.0pt\mathit{sc}}}}(A) denotes the category of semisimple perverse sheaves which are clean in the sense that they have no negligible subobjects, then for P,Q\in\operatorname{{P_{\hskip-2.0pt\mathit{sc}}}}(A) we have a unique decomposition

P*Q\;=\;P\circ Q\;\oplus\;P\bullet Q\quad\textnormal{with}\quad\begin{cases}P% \circ Q\;\in\;\operatorname{{P_{\hskip-2.0pt\mathit{sc}}}}(A),\\ P\bullet Q\;\in\;\operatorname{N}(A),\end{cases}

and \operatorname{{P_{\hskip-2.0pt\mathit{sc}}}}(A) equipped with the tensor product \circ is an inductive limit of Tannakian categories which lift the above Tannakian quotient categories. A similar lift exists in the non-semisimple case [KraemerSemiabelian, sect. 5], but working with semisimple perverse sheaves has the advantage that for these the decomposition theorem and the relative hard Lefschetz theorem holds by Kashiwara’s conjecture, see [DrinfeldKashiwara] [BKdeJong] [GaitsgoryDeJong] or [MochizukiAsymptotic] [SabbahPolarizable].

Now the objects we are interested in arise in the above Tannakian framework as follows. For P\in\operatorname{{P_{\hskip-2.0pt\mathit{sc}}}}(A) the symmetric group {\mathfrak{S}}_{n} acts on P^{n}=P\circ\cdots\circ P\in\operatorname{{P_{\hskip-2.0pt\mathit{sc}}}}(A) by permutation of the factors. Up to isomorphism the irreducible finite-dimensional rational representations V_{\alpha}\in\operatorname{Rep}_{\mathbb{Q}}({\mathfrak{S}}_{n}) are parametrized by the partitions \alpha of n, and by [DelCT, sect. 1.4]

P^{n}\;\;\cong\bigoplus_{\deg\alpha=n}S^{\alpha}(P)\otimes_{\mathbb{Q}}V_{% \alpha}\quad\textnormal{for the Schur functors}\quad S^{\alpha}:\;% \operatorname{{P_{\hskip-2.0pt\mathit{sc}}}}(A)\;\rightarrow\;\operatorname{{P% _{\hskip-2.0pt\mathit{sc}}}}(A).

For the partition \alpha=(n) we get the symmetric convolution powers S^{n}(P) which are the invariants under the symmetric group. Notice that if the perverse sheaf P underlies a mixed Hodge module or a polarized twistor module, then by the axioms in [KrWVanishing, sect. 5] the same will also hold for all the perverse sheaves S^{\alpha}(P) since the corresponding categories are {\mathbb{Q}}-linear pseudoabelian.

Unfortunately, very little is known about the perverse sheaves P^{n} and S^{\alpha}(P) in general. Put g=\dim A>0, and let U_{n}\subseteq A be an open dense subset over which the cohomology sheaves

{\mathcal{F}}_{n}(P)\;=\;{\mathcal{H}}^{-g}(P^{n})|_{U_{n}}\quad\;\textnormal{% and}\;\quad\tilde{{\mathcal{F}}}_{\alpha}\;=\;{\mathcal{H}}^{-g}(S^{\alpha}(P)% )|_{U_{n}}

are locally constant. The underlying monodromy representations are unknown even in the simplest nontrivial cases. The goal of this note is to determine the generic ranks r_{P}(n)=\operatorname{\mathit{rk}}({\mathcal{F}}_{n}(P)) and {\tilde{r}}_{P}(\alpha)=\operatorname{\mathit{rk}}(\tilde{{\mathcal{F}}}_{% \alpha}(P)). We will see that the generating series for the convolution powers and for the symmetric convolution powers

Z_{P}(t)\;=\;\sum_{n=1}^{\infty}\;r_{P}(n)\cdot t^{n}\quad\;\textnormal{and}\;% \quad\tilde{Z}_{P}(t)\;=\;\sum_{n=1}^{\infty}\;\tilde{r}_{P}(n)\cdot t^{n}

are rational functions of a very simple shape. To control the negligible summands we will assume that the spectrum {\mathscr{S}}(P) as defined in [KrWVanishing] is finite. Recall that {\mathscr{S}}(P) is the set of all characters

\varphi\;\in\;\Pi(A)\;=\;\operatorname{Hom}_{\mathbb{Z}}(\pi_{1}(A,0),{\mathbb% {C}}^{*})

such that the corresponding rank one local system L_{\varphi} satisfies \operatorname{H}^{i}(A,P\otimes_{\mathbb{C}}L_{\varphi})\neq 0 for some i\neq 0. The generic vanishing theorem of loc. cit. says that this spectrum is always a finite union of translates of algebraic subtori \Pi(B)\subset\Pi(A) defined by proper abelian quotient varieties A\twoheadrightarrow B, and our finiteness assumption means that only B=\{0\} occurs. This assumption is automatically satisfied if A is a simple abelian variety but also in many other applications, like for the perverse intersection cohomology sheaves on curves in lemma LABEL:lem:spectrum below or on ample divisors with only isolated singularities [KrE6, lemma 8.1]. This being said, we obtain the following result for the generating series for the generic ranks from above.

Theorem 1.1.

Let P\in\operatorname{{P_{\hskip-2.0pt\mathit{sc}}}}(A) be a semisimple clean perverse sheaf with a finite spectrum and with Euler characteristic \chi=\chi(P). Then

\displaystyle Z_{P}(t) \displaystyle\;=\;\frac{tf_{P}(t)}{(1-\chi t)^{g+1}}\quad\textnormal{\em for % some $f_{P}(t)\in{\mathbb{Z}}[t]$ of degree $\deg_{t}f_{P}(t)\leq g-1$},
\displaystyle{\tilde{Z}}_{P}(t) \displaystyle\;=\;\frac{t{\tilde{f}}_{P}(t)}{(1-t)^{2g+\chi}}\quad\textnormal{% \em for some ${\tilde{f}}_{P}(t)\in{\mathbb{Z}}[t]$ of degree $\deg_{t}{\tilde% {f}}_{P}(t)\leq 2g-2$},

and we have the functional equation


Notice that since we have introduced an extra factor t in the nominator of our rational functions, the constant term f_{P}(0)={\tilde{f}}_{P}(0) is equal to the generic rank of the cohomology sheaf {\mathcal{H}}^{-g}(P), which we will simply call the generic rank of P in what follows. The functional equation therefore implies that \deg_{t}{\tilde{f}}_{P}(t)=2g-2 iff the given perverse sheaf has support \mathrm{Supp}(P)=A. The examples below show that in general also \deg_{t}f_{P}(t)=g-1.

The proof of theorem 1.1 also gives explicit formulae for the arising polynomials in terms of Chern-MacPherson classes [MacPhersonChern], see theorem LABEL:thm:explicit and LABEL:thm:explicit_symmetric. We pass to the dual abelian variety \hat{A}={\mathit{Pic}}^{0}(A) via the Fourier transform since the latter replaces the convolution product by the intersection product [BeauvilleFourier] which is more convenient in explicit computations. The crucial input for the symmetric convolution powers is a result of Cappell, Maxim, Schürmann, Shaneson and Yokura [CappellSymmetric]. One may also express the other Schur functors via the characters of the symmetric group as follows, where \chi_{V_{\alpha}}:{\mathfrak{S}}_{n}\to{\mathbb{Z}} denotes the character of V_{\alpha}.

Theorem 1.2.

Let \alpha be a partition of n. For all clean semisimple P\in\operatorname{{P_{\hskip-2.0pt\mathit{sc}}}}(A) with finite spectrum one has

{\tilde{r}}_{P}(\alpha)\;=\;\frac{1}{n!}\sum_{\sigma\in{\mathfrak{S}}_{n}}\chi% _{V_{\alpha}}(\sigma)\cdot c_{P}(\sigma)

where the c_{P}(\sigma)\in{\mathbb{Z}} are integers given by the explicit formulae in theorem LABEL:thm:schur.

Like the character values, the integers c_{P}(\sigma) for a given P\in\operatorname{{P_{\hskip-2.0pt\mathit{sc}}}}(A) only depend on the conjugacy class of \sigma\in{\mathfrak{S}}_{n}. In the sequel we associate to any such class the cycle type


where \sigma_{i} denotes the length of the i^{\mathrm{th}} cycle in a decomposition of \sigma\in{\mathfrak{S}}_{n} into a product of disjoint cycles. For instance we have \ell(\sigma)=n iff \sigma is trivial. This being said, let us consider some examples. We begin with elliptic curves:

Theorem 1.3.

If g=1, then any clean semisimple P\in\operatorname{{P_{\hskip-2.0pt\mathit{sc}}}}(A) of generic rank r and Euler characteristic \chi=\chi(P) satisfies

f_{P}(t)\;=\;{\tilde{f}}_{P}(t)\;=\;r\quad\textnormal{\em and}\quad c_{P}(% \sigma)\;=\;r\,\chi^{\ell(\sigma)-1}\,\sum_{i=1}^{\ell(\sigma)}\sigma_{i}^{2}% \quad\textnormal{\em for}\quad\sigma\in{\mathfrak{S}}_{n}.

Note that the claim about the polynomials f_{P}(t) and {\tilde{f}}_{P}(t) already follows from the degree estimates in theorem 1.1 and from the above interpretation of the generic rank as the value of these polynomials at t=0. The formula for the numbers c_{P}(\sigma) is less obvious but will be explained later in corollary LABEL:cor:elliptic.

In higher dimensions the most interesting examples arise if P=\delta_{Z}\in\operatorname{P}(A) is the perverse intersection cohomology sheaf with support on a closed pure-dimensional subvariety Z\hookrightarrow A. Then P\in\operatorname{{P_{\hskip-2.0pt\mathit{sc}}}}(A) unless some irreducible component of Z is invariant under translations by a positive-dimensional abelian subvariety, and we put

f_{Z}(t)=f_{P}(t),\quad{\tilde{f}}_{Z}(t)={\tilde{f}}_{P}(t),\quad{\tilde{r}}_% {Z}(\alpha)={\tilde{r}}_{P}(\alpha)\quad\textnormal{and}\quad c_{Z}(\sigma)=c_% {P}(\sigma).

The simplest nontrivial instance occurs for Jacobian varieties, where the arising perverse sheaves are related to the theory of special linear series on curves [WeBN]. We denote by [h(s)]_{s^{g}} the coefficient of s^{g} in a polynomial h(s)\in{\mathbb{Z}}[s].

Theorem 1.4.

If A is the Jacobian variety of a smooth projective curve C\hookrightarrow A of genus g>1, then

f_{C}(t)\;=\;(g!-1)\,t^{g-1}\quad\textnormal{\em and}\quad{\tilde{f}}_{C}(t)\;% =\;t^{g-1}

and we have

c_{\hskip 0.2ptC}(\sigma)\;=\;\biggl{[}\,g!\prod_{i=1}^{\ell(\sigma)}\bigl{(}2% g-2+\sigma_{i}^{2}s\bigr{)}\,-\,\prod_{i=1}^{\ell(\sigma)}\bigl{(}2g-2-sq_{% \sigma_{i}}(s)\bigr{)}\,\biggr{]}_{s^{g}}

where q_{\sigma_{i}}(s)\in{\mathbb{Z}}[s] are polynomials given by the formula in definition LABEL:def:qpolynomial below.

The argument for Jacobian varieties generalizes directly to Prym-Tjurin varieties in the sense of [BL], see theorem LABEL:cor:prympowers. Another interesting case are the convolution powers of an ample divisor as in the following example [KrE6].

Example 1.5.

Let A be a general principally polarized abelian variety and \Theta\subset A a smooth ample divisor which defines the principal polarization. The convolution square \delta_{\Theta}\circ\delta_{\Theta} defines over some open dense subset a variation of Hodge structures whose fibres are the primitive cohomology of the intersections \Theta\cap(\Theta+x)\subset A for x\in A({\mathbb{C}}). These intersections have been studied a lot in the moduli theory of abelian varieties [BeauvillePrym] [DebTorelli]  [DebarreSchottkyUpdate] [IzadiGeometricStructure], and the higher convolutions \delta_{\Theta}\circ\cdots\circ\delta_{\Theta} provide a natural continuation of this topic. For g=4, examples LABEL:ex:g4convolution and LABEL:ex:g4symmetric will show

\displaystyle f_{\Theta}(t) \displaystyle\;=\;1829\,t^{3}-342\,t^{2}+58\,t,
\displaystyle{\tilde{f}}_{\Theta}(t) \displaystyle\;=\;52\,t^{5}+1292\,t^{4}+5049\,t^{3}+1292\,t^{2}+52\,t.

In particular, on some open dense subset the Hodge modules \delta_{\Theta}\circ\delta_{\Theta} and S^{2}(\delta_{\Theta}) define variations of Hodge structures of rank 58 and 52. Furthermore the alternating square S^{1,1}(\delta_{\Theta}) defines a variation of Hodge structures of rank {\tilde{r}}_{\Theta}(1,1)=58-52=6 whose stalks may be identified with the lattice E_{6} and whose monodromy group has index \leq 2 in \operatorname{Aut}(E_{6}); this is related to the Prym morphism [KrE6] [DonagiPrym].

2. Preliminary remarks and notations

The decomposition of tensor powers into Schur functors works in any {\mathbb{Q}}-linear symmetric monoidal pseudoabelian category [DelCT, sect. 1.2], so for any K\in\operatorname{D^{\mathit{b}}_{\mathit{c}}}(A) we may write

K^{*n}\;\cong\;\bigoplus_{\deg\alpha=n}\;S^{*\alpha}(K)\otimes_{\mathbb{Q}}V_{% \alpha}\qquad\textnormal{with}\qquad S^{*\alpha}(K)\in\operatorname{D^{\mathit% {b}}_{\mathit{c}}}(A).

If K=P is a clean semisimple perverse sheaf, then the decomposition theorem and the definition of the clean perverse sheaves P^{n} and S^{\alpha}(P) from the introduction imply that

\displaystyle P^{*n} \displaystyle\;\cong\;P^{n}\oplus P^{\bullet n}\hskip 52.0pt\textnormal{with}% \qquad P^{\bullet n}\in\operatorname{N}(A),
\displaystyle S^{*\alpha}(P) \displaystyle\;\cong\;S^{\alpha}(P)\oplus S^{\bullet\alpha}(P)\hskip 22.0pt% \textnormal{with}\qquad S^{\bullet\alpha}(P)\in\operatorname{N}(A).

So the data we are interested in becomes the difference of two parts: For \natural\in\{*,\bullet\} we put

\displaystyle r_{P}^{\natural}(n) \displaystyle\;=\;(-1)^{g}\,\chi_{\eta}(P^{\natural n}), \displaystyle Z_{P}^{\natural}(t) \displaystyle\;=\;\sum\nolimits_{n\geq 0}\,r_{P}^{\natural}(n)\cdot t^{n}
\displaystyle{\tilde{r}}_{P}^{\natural}(\alpha) \displaystyle\;=\;(-1)^{g}\,\chi_{\eta}(S^{\natural\alpha}(P)), \displaystyle{\tilde{Z}}_{P}^{\natural}(t) \displaystyle\;=\;\sum\nolimits_{n\geq 0}\,{\tilde{r}}_{P}^{\natural}(n)\cdot t% ^{n}

where \chi_{\eta}(-) denotes the Euler characteristic of the stalk cohomology at the generic point of the abelian variety. Then

Z_{P}\;=\;Z_{P}^{*}\;-\;Z_{P}^{\bullet},\qquad{\tilde{Z}}_{P}\;=\;{\tilde{Z}}_% {P}^{*}\;-\;{\tilde{Z}}_{P}^{\bullet}\qquad\textnormal{and}\qquad{\tilde{r}}_{% P}\;=\;{\tilde{r}}_{P}^{*}\;-\;{\tilde{r}}_{P}^{\bullet},

and the results from the introduction will be shown separately for each term.

To do this in a uniform way, let \operatorname{D}(A)\subseteq\operatorname{D^{\mathit{b}}_{\mathit{c}}}(A) be the full additive subcategory of all sheaf complexes that decompose as a direct sum \bigoplus_{n\in{\mathbb{Z}}}P_{n}[n] for semisimple perverse sheaves P_{n}\cong P_{-n} with finite spectrum. From the decomposition theorem and the relative hard Lefschetz theorem one sees that this subcategory is stable under the convolution product. Consider the associated Grothendieck ring


where the underlying Grothendieck group is taken only with respect to direct sums, not distinguished triangles: We keep the information on the degrees in which a complex sits. To compute the above terms with the superscript \natural=\bullet resp. \natural=* we consider ring homomorphisms

\Addop@@=0.5em@R=1.8em@C=3.5em{&&&\operatorname{K}(A){}{}{}{}{}{}{}{}{}{}% \xy@@ix@{{\hbox{}}}}
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