Smith theory and irreducible holomorphic symplectic manifolds

By Samuel Boissière, Marc A. Nieper-Wisskirchen, A. Sarti. 2012

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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.

Subfields: Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/83e54d6ce4b36682495e52e6ac067585d78ce742