Geodesic
Twisted cohomology of the Hilbert schemes of points on surfaces
Documenta mathematica, Tome 14 (2009), pp. 749-770.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: 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.
Classification : 14C05, 14C15
Keywords: Hilbert schemes of points on surfaces, rational cohomology ring, locally constant systems, generalised Kummer varieties 
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     author = {Nieper-Wi{\ss}kirchen, Marc A.},
     title = {Twisted cohomology of the {Hilbert} schemes of points on surfaces},
     journal = {Documenta mathematica},
     pages = {749--770},
     publisher = {mathdoc},
     volume = {14},
     year = {2009},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DOCMA_2009__14__a2/}
}
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Nieper-Wißkirchen, Marc A. Twisted cohomology of the Hilbert schemes of points on surfaces. Documenta mathematica, Tome 14 (2009), pp. 749-770. http://geodesic.mathdoc.fr/item/DOCMA_2009__14__a2/