On the top-weight rational cohomology of 𝒜g

Brandt, Madeline; Bruce, Juliette; Chan, Melody; Melo, Margarida; Moreland, Gwyneth; Wolfe, Corey

doi:10.2140/gt.2024.28.497

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On the top-weight rational cohomology of $\mathcal{A}_g$

Madeline Brandt, Juliette Bruce, Melody Chan, Margarida Melo, Gwyneth Moreland and Corey Wolfe

Geometry & Topology 28 (2024) 497–538

DOI: 10.2140/gt.2024.28.497

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 and Soulé. In this way, our results make a precise connection between the rational cohomology of ${Sp}_{2 g} (ℤ)$ and ${GL}_{g} (ℤ)$ . Our computations also 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.

Keywords

top-weight cohomology, moduli space of abelian varieties, toroidal compactifications

Mathematical Subject Classification

Primary: 14K10, 14T90

Secondary: 14F25

References

Publication

Received: 12 February 2021

Revised: 7 June 2022

Accepted: 8 July 2022

Published: 13 March 2024

Proposed: Mladen Bestvina

Seconded: Mark Gross, Dan Abramovich

Authors

Madeline Brandt
Department of Mathematics Brown University Providence, RI United States

Juliette Bruce
Department of Mathematics University of California, Berkeley Berkeley, CA United States	Department of Mathematics Brown University Providence, RI United States

Melody Chan
Department of Mathematics Brown University Providence, RI United States

Margarida Melo
Department of Mathematics and Physics Università Roma Tre Rome Italy

Gwyneth Moreland
Department of Mathematics Harvard University Cambridge, MA United States	Department of Mathematics, Statistics and Computer Science University of Illinois Chicago Chicago, IL United States

Corey Wolfe
Department of Mathematics Tulane University New Orleans, LA United States

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Volume 28, issue 2 (2024)