The Rational Cohomology Ring of the Moduli Space of Abelian $3$-folds

Abstract

The rational cohomology ring of A_3, the moduli space of abelian 3-folds is computed. This is isomorphic to the the rational cohomology ring of the group Sp_3(Z) of 6x6 integral symplectic matrices. The main ingredients in the computation are (1) Looijenga's computation of the rational cohomology ring of M_3, the moduli space of smooth projective curves of genus 3, and (2) the stratified Morse theory of Goresky and MacPherson, which we use to compute the homology of the jacobian locus (in the rank 3 Siegel upper half plane), or equivalently of the extended Torelli group in genus 3. In the revised version, we also compute the rational cohomology of the Satake compactification of A_3.