Masterclass Lecture by Orsola Tommasi II

Lecture in Masterclass: Cohomology of arithmetic groups by Orsola Tommasi (Chalmers University of Technology and University of Gothenburg)

Title: Cohomological stabilization of toroidal compactifications of A_g

Abstract: By a classical result of Borel, the cohomology of the symplectic group Sp(2g,Z) stabilizes in degree k<g. This can be interpreted geometrically as a stabilization result for the rational cohomology of A_g, the moduli space of complex g-dimensional principally polarized abelian varieties. Work of Charney and Lee provides an analogous result for the stable cohomology of the minimal compactification of A_g, the Baily-Borel-Satake compactification.

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

This is joint work with Sam Grushevsky and Klaus Hulek.