On the Top-Weight Rational Cohomology of $A_g$

Abstract

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-Soulé. Our computations 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.

Realizability of tropical pluri-canonical divisors

“…In the recent and much-celebrated work [CGP21] the authors computed the top weight cohomology of M g from the reduced rational cohomology of the link of M trop g . The same technique was shortly after applied to compute the top weight cohomology for some instances of the moduli space of abelian varieties A g in [BBCMMW21]. In both cases it is vital to identify the tropical moduli space with the boundary complex of the classical moduli space (see e.g.…”