The abundance and SYZ conjectures in families of hyperkahler manifolds

TL;DR

This paper proves the SYZ conjecture for certain hyperkahler manifolds by establishing deformation invariance of semiampleness using a new Teichmuller space framework.

Contribution

It introduces a Teichmuller space for pairs (M,L) and proves a global Torelli theorem to show deformation invariance of semiampleness in hyperkahler manifolds.

Findings

Proved SYZ conjecture under deformation assumptions.

Established a global Torelli theorem for the new Teichmuller space.

Demonstrated deformation invariance of semiampleness.

Abstract

Let $L$ be a holomorphic line bundle on a hyperkahler manifold $M$, with $c_1(L)$ nef and not big. SYZ conjecture predicts that $L$ is semiample. We prove that this is true, assuming that $(M,L)$ has a deformation $(M',L')$ with $L'$ semiample. We introduce a version of the Teichmuller space that parametrizes pairs $(M,L)$ up to isotopy. We prove a version of the global Torelli theorem for such Teichmuller spaces and use it to deduce the deformation invariance of semiampleness.