Stable cohomology of the perfect cone toroidal compactification of ${\mathcal A}_g$

TL;DR

This paper proves that the cohomology of the perfect cone compactification of the moduli space of abelian varieties stabilizes near the top degree, is purely algebraic, and provides explicit computations up to degree 13.

Contribution

It establishes the stabilization and algebraicity of cohomology for the perfect cone compactification and related spaces, with explicit calculations up to degree 13.

Findings

Cohomology stabilizes near the top degree.

Stable cohomology is purely algebraic.

Explicit generators identified up to degree 8.

Abstract

We show that the cohomology of the perfect cone (also called first Voronoi) toroidal compactification of the moduli space of complex principally polarized abelian varieties stabilizes, in close to the top degree. Moreover, we show that this stable cohomology is purely algebraic, and we compute it in degree up to 13. Our explicit computations and stabilization results apply in greater generality to various toroidal compactifications and partial compactifications, and in particular we show that the cohomology of the matroidal partial compactification stabilizes (in low degree). For degree up to 8, we describe explicitly the generators of the cohomology. We also discuss various approaches to computing all of the stable cohomology in arbitrary degree.