A global Torelli theorem for hyperkahler manifolds

TL;DR

This paper proves a Torelli theorem for hyperkähler manifolds, establishing a deep link between their complex structures and period domains, with implications for their moduli spaces and mapping class groups.

Contribution

It introduces a birational Teichmüller space and demonstrates an isomorphism with a period space, extending Torelli theorems to hyperkähler manifolds with new techniques.

Findings

Mapping class group is commensurable to an arithmetic subgroup in SO(3, b_2-3)

Period map is an isomorphism between birational Teichmüller space and a period space

Torelli theorem identifies moduli space components with quotients of period spaces

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 hypekahler manifold $M$, showing that it is commensurable to an arithmetic subgroup in SO(3, b_2-3). 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 an isomorphism of the birational Teichmuller space and the corresponding period space $SO(b_2-3, 3)/SO(2)\times SO(b_2 -3, 1)$. We use this result to obtain a Torelli theorem identifying any connected component of 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 Hodge-theoretic birational Torelli theorem (for other examples of hyperkahler manifolds the Hodge-theoretic Torelli theorem is known to be false).