Mapping class group and a global Torelli theorem for hyperkähler manifolds

Abstract

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 hyperkähler manifold $M$, showing that it is commensurable to an arithmetic lattice in $SO(3,b_2-3)$. 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. We show that the period map gives the isomorphism between connected components of the birational Teichmüller space and the corresponding period space $SO(b_2-3,3)/SO\left(2\right)\times SO(b_2-3,1)$. We use this result to obtain a Torelli theorem identifying each connected component of the birational moduli space with a quotient of a period space by an arithmetic group. 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 Hodge-theoretic birational Torelli theorem (for other examples of hyperkähler manifolds, the Hodge-theoretic Torelli theorem is known to be false).