Mapping class group and a global Torelli theorem for hyperkähler manifolds

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 hypekähler manifold M , showing that it is commensurable to an arithmetic lattice in SO(3, b2 − 3). A Teichmüller space of M is a space of complex structures on M up to isotopies. We define a birational Teichmüller space by identifying certain points corresponding to bimeromorphically equivalent manifolds, and show that the period map gives the isomorphism of the birational Teichmüller 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. When M is a Hilbert scheme of n points on a K3 surface, with n − 1 a prime power, our Torelli theorem implies the usual Hodgetheoretic birational Torelli theorem (for other examples of hyperkähler manifolds the Hodge-theoretic Torelli theorem is known to be false).