Smith theory and irreducible holomorphic symplectic manifolds

TL;DR

This paper investigates the cohomological effects of prime order automorphisms on irreducible holomorphic symplectic manifolds, establishing relations between fixed locus cohomology and invariant lattice properties.

Contribution

It provides a new formula linking the mod p cohomology of fixed loci to the invariant lattice structure in the second cohomology of these manifolds.

Findings

Relation between mod p cohomology of fixed locus and invariant lattice

Formula connecting fixed locus cohomology dimension with lattice rank and discriminant

Application to manifolds deformation equivalent to Hilbert schemes of K3 surfaces

Abstract

We study the cohomological properties of the fixed locus $X^G$ of an automorphism group $G$ of prime order $p$ acting on a variety $X$ whose integral cohomology is torsion-free. We obtain an precise relation between the mod $p$ cohomology of $X^G$ and natural invariants for the action of $G$ on the integral cohomology of $X$. We apply these results to irreducible holomorphic symplectic manifolds of deformation type of the Hilbert scheme of two points on a K3 surface: the main result of this paper is a formula relating the dimension of the mod $p$ cohomology of $X^G$ with the rank and the discriminant of the invariant lattice in the second cohomology space with integer coefficients of $X$.