On the transcendental lattices of Hyperk\"ahler manifolds

TL;DR

This paper introduces a new perspective on Hyper-K"ahler manifolds via Hodge structures, providing criteria to identify when they are related to moduli spaces of sheaves, with special focus on O'Grady type manifolds.

Contribution

It defines Hyper-K"ahler manifolds induced by K3-type Hodge structures and establishes lattice-theoretic criteria for their birationality to moduli spaces, especially analyzing O'Grady type cases.

Findings

Lattice-theoretic criteria for birationality to moduli spaces

Different behaviors observed in O'Grady type manifolds

Characterization of Hyper-K"ahler manifolds induced by K3 surfaces

Abstract

We introduce the notion of a Hyper-K\"{a}hler manifold $X$ induced by a Hodge structure of K3-type. We explore this notion for the known deformation types of hyper-K\"{a}hler manifolds studying those that are induced by a K3 or abelian surface, giving lattice-theoretic criteria to decide whether or not they are birational to a moduli space of sheaves over said surface. We highlight the different behaviors we find for the particular class of hyper-K\"{a}hler manifolds of O'Grady type.