Oberseminar WiSe 2009/10 Zusammenfassung des Vortrags von O. Tommasi
Towards a computation of the cohomology of the moduli space of abelian fourfolds
The cohomology of moduli spaces of principally polarized abelian varieties is known only in dimension ≤3. In these cases, the proof of the result relies on the fact that the Torelli map is dominant, i.e., a general abelian variety is the jacobian of a curve. This enables one to deduce cohomological information on the moduli space of abelian varieties from results on the moduli space of curves. In dimension 4, the Torelli map is no longer dominant and its image is a divisor in the moduli space A4 of abelian fourfolds. Nevertheless, one can pose the problem of how much of the cohomology of A4 can be obtained from the knowledge of the cohomology of the moduli space of genus 4 curves and of moduli spaces of abelian varieties of lower dimension. We will investigate this problem for the cohomology with rational coefficients of the second Voronoi compactification of A4, a toroidal compactification which is particularly interesting for geometric reasons. We will explain why this approach should allow us to describe the cohomology of this space in all degrees, with the exception of certain intermediate ones.
This is joint work (in progress) with Klaus Hulek.