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arXiv · 1204.4118

Smith theory and irreducible holomorphic symplectic manifolds

Samuel Boissiere·Marc Nieper-Wisskirchen·Alessandra Sarti
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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 $X$.

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Samuel Boissiere, Marc Nieper-Wisskirchen, Alessandra Sarti. 2012-07-06. Smith theory and irreducible holomorphic symplectic manifolds. https://doi.org/10.1112/jtopol%2Fjtt002

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