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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

We compute the top-weight rational cohomology of 𝒜g for g = 5, 6 and 7, and we give some vanishing results for the top-weight rational cohomology of 𝒜8,𝒜9 and 𝒜10. When g = 5 and g = 7, we exhibit nonzero cohomology groups of 𝒜g in odd degree, thus answering a question highlighted by Grushevsky. Our methods develop the relationship between the top-weight cohomology of 𝒜g 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 2g() and GL g(). Our computations also give natural candidates for compactly supported cohomology classes of 𝒜g in weight 0 that produce the stable cohomology classes of the Satake compactification of 𝒜g in weight 0, under the Gysin spectral sequence for the latter space.

top-weight cohomology, moduli space of abelian varieties, toroidal compactifications
Mathematical Subject Classification
Primary: 14K10, 14T90
Secondary: 14F25
Received: 12 February 2021
Revised: 7 June 2022
Accepted: 8 July 2022
Published: 13 March 2024
Proposed: Mladen Bestvina
Seconded: Mark Gross, Dan Abramovich
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
UniversitĂ  Roma Tre
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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