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arXiv · 2012.02892

On the Top-Weight Rational Cohomology of $A_g$

Madeline Brandt·Juliette Bruce·Melody Chan·Margarida Melo·Gwyneth Moreland·Corey Wolfe
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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.

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Madeline Brandt, Juliette Bruce, Melody Chan, Margarida Melo, Gwyneth Moreland, Corey Wolfe. 2020-12-04. On the Top-Weight Rational Cohomology of $A_g$. https://arxiv.org/abs/2012.02892

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