On the fixed locus of the antisymplectic involution of an EPW cube

TL;DR

This paper investigates the fixed locus of an anti-symplectic involution on EPW cubes, revealing it as a rigid Lagrangian submanifold through degeneration techniques, advancing understanding of hyper-K"ahler geometry.

Contribution

It provides a detailed geometric analysis of the fixed locus of the involution on EPW cubes, establishing its rigidity as a Lagrangian submanifold, which is a novel insight in hyper-K"ahler geometry.

Findings

The fixed locus is a rigid atomic Lagrangian submanifold.

Degeneration methods are used to analyze singular degenerations of EPW cubes.

The study advances understanding of involutions on hyper-K"ahler varieties.

Abstract

EPW cubes are polarized hyper-K\"ahler varieties of K$3^{[3]}$-type that carry an anti-symplectic involution. We study the geometry of the fixed locus $\sW_A$ of this involution and prove that it is a \emph{rigid} atomic Lagrangian submanifold. Our proof is based on a detailed description of certain singular degenerations of EPW cubes and the degeneration methods of Flappan--Macr\`i--O'Grady--Sacc\`a.