Equivariant cohomology of moduli spaces of genus three curves with level two structure

Abstract

We compute cohomology of the moduli space of genus three curves with level two structure and some related spaces. In particular, we determine the cohomology groups of the moduli space of plane quartics with level two structure as representations of the symplectic group on a six dimensional vector space over the field of two elements. We also make the analogous computations for some related spaces such as moduli spaces of genus three curves with a marked points and strata of the moduli space of Abelian differentials of genus three. BackgroundWe work over over the field of complex numbers.2.1. Level structures. Let C be a smooth and irreducible curve of genus g and let Jac(C) denote its Jacobian. For any positive integer n there is an isomorphismwhere Jac(C)[n] denotes the subgroup of n-torsion elements in Jac(C). A symplectic level n structure on C is an ordered basis (D 1 , . . . , D 2g ) of Jac(C)[n] such that the Weil pairing has matrix of the form 0 I g −I g 0

The equivariant Euler characteristic of $\mathcal{A}3[2]$

“…Let A g [2] denote the moduli space of principally polarized abelian varieties of dimension g together with a full level 2 structure. Similarly, let M g [2] denote the moduli space of smooth curves of genus g together with a full level 2 structure. Note that we are considering these as coarse moduli spaces.…”

“…Note that we are considering these as coarse moduli spaces. The two types of moduli spaces are connected through the Torelli morphism t g : M g [2] → A g [2] sending a smooth curve to its Jacobian. There is an action of Sp 2 (2g), the symplectic group of degree 2g over the field of two elements, on A g [2] and M g [2] via its action on the level 2 structure (for some more details see for instance [2]).…”

“…For g ≤ 2, M g [2] is isomorphic to the locus H g [2] consisting of hyperelliptic curves. The space H g [2] can in turn be described as a disjoint union of copies of M 0,2g+2 (see [6], [10] and [11]), the moduli space of smooth genus 0 curves together with 2g + 2 marked points.…”

“…For g ≤ 2, M g [2] is isomorphic to the locus H g [2] consisting of hyperelliptic curves. The space H g [2] can in turn be described as a disjoint union of copies of M 0,2g+2 (see [6], [10] and [11]), the moduli space of smooth genus 0 curves together with 2g + 2 marked points. The cohomology of M 0,2g+2 , together with the action of the symmetric group S 2g+2 , can (because of purity) be computed using counts of points over finite fields (see for instance [4] and [9]).…”

“…The cohomology of M 0,2g+2 , together with the action of the symmetric group S 2g+2 , can (because of purity) be computed using counts of points over finite fields (see for instance [4] and [9]). In Section 5.1 respectively Section 5.2 below, we compute in this way the Sp 2 (2)-action and Hodge structure of the cohomology of M 1 [2] ∼ = A 1 [2] respectively the Sp 2 (4)-action and Hodge structure of the cohomology of M 2 [2]. By adding the complement of t 2 (M 2 [2]) inside A 2 [2] consisting of products of elliptic curves we also compute the Sp 2 (4)equivariant weighted Euler characteristic of A 2 [2].…”