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Oberseminar SoSe 2011 Zusammenfassung des Vortrags von Misha Verbitsky (Moskau, RUS)

Global Torelli theorem for hyperkaehler manifolds and the mapping class group

A mapping class group of an oriented manifold is a quotient of its diffeomorphism group by the isotopies. We compute a mapping class group of a hypekahler manifold M, showing that it is commensurable to an arithmetic lattice in SO (3, b2 − 3). A Teichmuller space of M is a space of complex structures on M up to isotopies. We define a birational Teichmuller space by identifying certain points corresponding to bimeromorphically equivalent manifolds, and show that the period map gives the isomorphism of the birational Teichmuller space and the corresponding period space SO (b2 − 3,3) / SO (2) × SO (b2 − 3,1). We use this result to obtain a Torelli theorem identifying the birational moduli space with a quotient of a period space by an arithmetic subgroup.