Symplectic involutions of holomorphic symplectic fourfolds

TL;DR

This paper investigates symplectic involutions on certain holomorphic symplectic fourfolds, establishing fixed locus properties and conjecturing a specific fixed point and surface configuration supported by examples.

Contribution

It introduces a conjecture on the fixed locus structure of symplectic involutions on holomorphic symplectic fourfolds and provides evidence through various examples.

Findings

Fixed locus contains at least 12 isolated points and 1 surface.

Conjecture: fixed locus has 28 isolated points and 1 K3 surface.

Evidence provided from examples like Hilbert schemes and Fano varieties.

Abstract

Let X be a holomorphic symplectic fourfold such that b_2=23 and i a symplectic involution of X . The fixed locus F of i is a smooth symplectic submanifold of X; we show that F contains at least 12 isolated points and 1 smooth surface. We conjecture that F is made of 28 isolated fixed points and 1 K3 surface and we provide evidences for the conjecture in some examples, as the Hilbert scheme of a K3 surface, the Fano variety of a cubic in P^5 and the double cover of an EPW sextic.