Double EPW-sextics with actions of A7 and irrational GM threefolds

TL;DR

This paper constructs highly symmetric hyper-K"ahler fourfolds with A7 actions as double EPW sextics and uses these to explicitly build a family of irrational Gushel-Mukai threefolds, advancing understanding of symmetries in complex geometry.

Contribution

It introduces new examples of symmetric hyper-K"ahler fourfolds with A7 actions and links them to irrational Gushel-Mukai threefolds through explicit constructions.

Findings

Construction of hyper-K"ahler fourfolds with A7 symmetry

Explicit family of irrational Gushel-Mukai threefolds

Identification of double EPW sextics as key geometric objects

Abstract

We construct two examples of projective hyper-K\"ahler fourfolds of K3[2]-type with an action of the alternating group A7, making them some of the most symmetric hyper-K\"ahler fourfolds. They are realized as so called double EPW sextics and this allows us to construct an explicit family of irrational Gushel-Mukai threefolds.