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Oberseminar SoSe 2011 Zusammenfassung des Vortrags von Valery Alexeev (Univ. Georgia, USA)

On the three compactifications of Siegel space

The moduli space Ag of abelian varieties has three classical toroidal compactifications: (1) perfect, (2) 2nd Voronoi, and (3) Igusa blowup, each with its own distinct geometric meaning. It is an interesting problem to understand exactly how these compactifications are related.

I will show that (1) and (2) are isomorphic in a neighborhood of the image of a regular map from the Deligne-Mumford's moduli space Mg, and that the rational map Mg - - -> Ag for (3) is not regular for g >8. This is a joint work with Adrian Brunyate.