On the Top-Weight Rational Cohomology of $\mathcal{A}_g$

Abstract

We compute the top-weight rational cohomology of $\mathcal{A}_g$ for $g=5$, $6$, and $7$, and we give some vanishing results for the top-weight rational cohomology of $\mathcal{A}_8, \mathcal{A}_9,$ and $\mathcal{A}_{10}$. When $g=5$ and $g=7$, we exhibit nonzero cohomology groups of $\mathcal{A}_g$ in odd degree, thus answering a question highlighted by Grushevsky. Our methods develop the relationship between the top-weight cohomology of $\mathcal{A}_g$ 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 $\mathcal{A}_g$ in weight $0$ that produce the stable cohomology classes of the Satake compactification of $\mathcal{A}_g$ in weight $0$, under the Gysin spectral sequence for the latter space.Comment: 33 pages, 3 figure