On the Top-Weight Rational Cohomology of $A_g$

TL;DR

This paper computes the top-weight rational cohomology of the moduli space of principally polarized abelian varieties for specific genera, revealing new nonzero cohomology groups and establishing links with tropical geometry and Voronoi complexes.

Contribution

It provides explicit computations of top-weight cohomology for $A_g$ at certain genera and introduces methods connecting tropical geometry with classical cohomology.

Findings

Nonzero cohomology in odd degrees for $A_5$ and $A_7$

Vanishing results for $A_8$, $A_9$, and $A_{10}$

Candidates for compactly supported cohomology classes in weight 0

Abstract

We compute the top-weight rational cohomology of $A_g$ for $g=5$, $6$, and $7$, and we give some vanishing results for the top-weight rational cohomology of $A_8, A_9,$ and $ A_{10}$. When $g=5$ and $g=7$, we exhibit nonzero cohomology groups of $A_g$ in odd degree, thus answering a question highlighted by Grushevsky. Our methods develop the relationship between the top-weight cohomology of $A_g$ and the homology of the link of the moduli space of principally polarized tropical abelian varieties of rank $g$. To compute the latter we use the Voronoi complexes used by Elbaz-Vincent-Gangl-Soul\'e. Our computations give natural candidates for compactly supported cohomology classes of $A_g$ in weight $0$ that produce the stable cohomology classes of the Satake compactification of $A_g$ in weight $0$, under the Gysin spectral sequence for the latter space.