The stable cohomology of the Satake compactification of $\mathcal{A}_g$

TL;DR

This paper provides an algebraic proof of the stabilization of rational cohomology for the Satake-Baily-Borel compactification of moduli spaces of abelian varieties, revealing the mixed Hodge structure as impure.

Contribution

It offers a simpler algebraic proof of Charney and Lee's theorem, incorporating the mixed Hodge structure analysis.

Findings

Cohomology stabilizes as g increases

The mixed Hodge structure is impure

Provides an algebraic approach to the cohomology stability

Abstract

Charney and Lee have shown that the rational cohomology of the Satake-Baily-Borel compactification the moduli space of principally polarized abelian varieties of dimension g stabilizes as g grows and they computed this stable cohomology as a Hopf algebra. We give a relatively simple algebro-geometric proof of their theorem that also takes into account the mixed Hodge structure that is present here. We find the latter to be impure.