Talk by prof. Orsola Tommasi

[Ugo Bruzzo]

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SISSA GEOMETRY SEMINARS
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Prof. ORSOLA TOMMASI (Leibniz Universität Hannover)
will give a talk on

COHOMOLOGICAL STABILITY FOR COMPACTIFICATIONS 
OF THE MODULI SPACE OF ABELIAN VARIETIES

on Monday, April 11th, at 4 pm in Room 005 (SISSA Building A).

Abstract:

Abelian varieties are projective varieties endowed with a group structure on their set of points. Isomorphism classes of (principally polarized) abelian varieties of dimension g are parametrized by the moduli space A_g. However, the space A_g is not compact, and for many applications it is useful to work with a compactification of A_g. There are several known families of such compactifications, including the Satake compactification and the toroidal compactifications constructed by Mumford and his collaborators.

Studying the topology of A_g is a way to get information about the behaviour of families of abelian varieties. In this talk, we consider one of the most important topological invariants, namely, the cohomology groups H^k with rational coefficients. For A_g, the cohomology group H^k is independent of g for k<g by a classical theorem of Borel. In other words, the group H^k stabilizes for large g.  An analogous result holds for the Satake compactification by work of Charney and Lee. In joint work with Sam Grushevsky and Klaus Hulek, we investigated the stabilization behaviour for toroidal compactifications, proving stability results for the perfect cone compactification and of the matroidal partial compactification.