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Hilbert Schemes of Points on Surfaces



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749 Documenta Math Twisted Cohomology of the Hilbert Schemes of Points on Surfaces Marc A Nieper Wi kirchen1 Received February 1 2008 Revised April 23 2009 Communicated by Thomas Peternell Abstract We calculate the cohomology spaces of the Hilbert schemes of points on surfaces with values in local systems For that purpose we generalise I Grojnoswki s and H Nakajima s description of the ordinary cohomology in terms of a Fock space representation to the twisted case We make further non trivial generalisations of M Lehn s work on the action of the Virasoro algebra to the twisted and the non projective case Building on work by M Lehn and Ch Sorger we then give an explicit description of the cup product in the twisted case whenever the surface has a numerically trivial canonical divisor We formulate our results in a way that they apply to the projective and non projective case in equal measure As an application of our methods we give explicit models for the cohomology rings of the generalised Kummer varieties and of a series of certain even dimensional Calabi Yau manifolds 2000 Mathematics Subject Classification 14C05 14C15 Keywords and Phrases Hilbert schemes of points on surfaces rational cohomology ring locally constant systems generalised Kummer varieties 1 The author would like to thank the Max Planck Institute for Mathematics in Bonn for its hospitality and support during the preparation of this paper He would also like to thank the referee for some very helpful suggestions Documenta Mathematica 14 2009 749 770 750 Marc A Nieper Wi kirchen Contents 1 Introduction and results 2 The Fock space description 3 The Virasoro algebra in the twisted case 4 The boundary operator 5 The ring structure 6 The generalised Kummer varieties References 750 754 758 760 762 767 769 1 Introduction and results Let X be a quasi projective smooth surface over the complex numbers We denote by X n the Hilbert scheme of n points on X parametrising zerodimensional subschemes of X of length n



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