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Smith theory and irreducible holomorphic symplectic manifolds

We study the cohomological properties of the fixed locus XG of an automorphism group G of prime order p acting on a variety X whose integral cohomology is torsion‐free. We obtain a precise relation between the mod p cohomology of XG 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 XG with the rank and the discriminant of the invariant lattice in the second cohomology space with integer coefficients of X.