Teichm\"uller spaces and Torelli theorems for hyperk\"ahler manifolds

TL;DR

This paper clarifies the correct Torelli theorem for hyperk"ahler manifolds, providing a precise statement, a concise proof, and revisiting some of its implications with new shorter proofs.

Contribution

It corrects previous misunderstandings of the Torelli theorem for hyperk"ahler manifolds and offers a streamlined proof and analysis of its consequences.

Findings

Counterexamples to the original Torelli statement for hyperk"ahler manifolds

A corrected version of the Torelli theorem

Shorter proofs of some consequences

Abstract

Kreck and Yang Su recently gave counterexamples to a version of the Torelli theorem for hyperk\"ahlerian manifolds as stated by Verbitsky. The initial purpose of this document (which was prepared for a seminar talk) was to extract the correct statement and to give a short proof of it. We also revisit a few of its consequences, some of which are given new (shorter) proofs.