Smith theory and irreducible holomorphic symplectic manifolds

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 $X$.

Some results improved. In particular, the main result holds for order three automorphisms

Cited by in corpus (6)

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• Symplectic involutions of $K{3}^{[n]}$ type and Kummer $n$ type manifolds
• Computing Cup-Products in integral cohomology of Hilbert schemes of points on K3 surfaces
• Symmetric Powers of Symmetric Bilinear Forms, Homogeneous Orthogonal Polynomials on the Sphere and an Application to Compact Hyperkähler Manifolds