Hyperk\"ahler manifolds

TL;DR

This paper provides an overview of hyperk"ahler manifolds, focusing on their geometric, automorphic, and lattice-theoretic properties, especially for projective deformations of K3 surface Hilbert schemes.

Contribution

It offers a comprehensive survey of results on projective embeddings, automorphisms, moduli spaces, and period maps for hyperk"ahler manifolds, including explicit descriptions of period map images.

Findings

Analysis of automorphism groups of hyperk"ahler manifolds

Descriptions of moduli spaces and period domains

Explicit characterization of period map images for polarized cases

Abstract

The aim of these notes is to acquaint the reader with important objects in complex algebraic geometry: K3 surfaces and their higher-dimensional analogs, hyperk\"ahler manifolds. These manifolds are interesting from several points of view: dynamical (some have interesting automorphism groups), arithmetical (although we will not say anything on this aspect of the theory), and geometric. It is also one of those rare cases where the Torelli theorem allows for a powerful link between the geometry of these manifolds and lattice theory. We do not prove all the results that we state. Our aim is more to provide, for specific families of hyperk\"ahler manifolds (which are projective deformations of punctual Hilbert schemes of K3 surfaces), a panorama of results about projective embeddings, automorphisms, moduli spaces, period maps and domains, rather than a complete reference guide. These results are mostly not new, except perhaps those of Appendix B (written with E. Macr\`i), where we give an explicit description of the image of the period map for these polarized manifolds.