On the top-weight rational cohomology of 𝒜g

Brandt, Madeline¹ ; Bruce, Juliette² ; Chan, Melody¹ ; Melo, Margarida³ ; Moreland, Gwyneth⁴ ; Wolfe, Corey⁵

¹ Department of Mathematics, Brown University, Providence, RI, United States
² Department of Mathematics, University of California, Berkeley, Berkeley, CA, United States, Department of Mathematics, Brown University, Providence, RI, United States
³ Department of Mathematics and Physics, Università Roma Tre, Rome, Italy
⁴ Department of Mathematics, Harvard University, Cambridge, MA, United States, Department of Mathematics, Statistics and Computer Science, University of Illinois Chicago, Chicago, IL, United States
⁵ Department of Mathematics, Tulane University, New Orleans, LA, United States

Geometry & topology, Tome 28 (2024) no. 2, pp. 497-538

Résumé

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(ℤ) and GL ⁡ g(ℤ). 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.

Détail

DOI : 10.2140/gt.2024.28.497

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

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