Orsola Tommasi: Cohomological stabilization of toroidal compactifications of A_g

Principally polarized abelian varieties of dimension g and their moduli space A_g are basic objects in algebraic geometry. However, determining the cohomology of A_g is an open problem in general. In small degree k<g, the cohomology of A_g is freely generated by the odd Chern classes of the Hodge bundle by a classical result of Borel. Work of Charney and Lee provides an analogous result for the stable cohomology of the minimal compactification of A_g, the Satake compactification.

For most geometric applications, it is more natural to consider toroidal compactifications of A_g instead. In this talk, we will explain our stability results for the perfect cone compactification and the matroidal partial compactification and discuss the structure of the stable cohomology groups that arise in this way.

This is joint work with Sam Grushevsky and Klaus Hulek.