# On the period map for polarized hyperk\"ahler fourfolds

## TL;DR

This paper investigates the period map for polarized hyperk"ahler fourfolds, showing its image's complement is a union of explicit Heegner divisors and characterizing when these fourfolds are isomorphic to known types.

## Contribution

It explicitly describes the complement of the period map image and identifies conditions for hyperk"ahler fourfolds to be isomorphic to Hilbert squares or double EPW sextics.

## Key findings

- The period map is an open embedding with a complement of explicit Heegner divisors.
- Infinitely many Heegner divisors correspond to fourfolds isomorphic to Hilbert squares or double EPW sextics.
- Automorphism groups of certain hyperk"ahler fourfolds are determined.

## Abstract

This is an improved version of the eprint previously entitled "Unexpected isomorphisms between hyperk\"ahler fourfolds." We study smooth projective hyperk\"ahler fourfolds that are deformations of Hilbert squares of K3 surfaces and are equipped with a polarization of fixed degree and divisibility. They are parametrized by a quasi-projective irreducible 20-dimensional moduli space and Verbitksy's Torelli theorem implies that their period map is an open embedding. Our main result is that the complement of the image of the period map is a finite union of explicit Heegner divisors that we describe. We also prove that infinitely many Heegner divisors in a given period space have the property that their general points correspond to fourfolds which are isomorphic to Hilbert squares of a K3 surfaces, or to double EPW sextics. In two appendices, we determine the groups of biregular or birational automorphisms of various projective hyperk\"ahler fourfolds with Picard number 1 or 2.