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Computing cup products in integral cohomology of Hilbert schemes of points on K3 surfaces

Part of: Computational aspects in algebraic geometry Homology and cohomology theories Combinatorics Surfaces and higher-dimensional varieties

Published online by Cambridge University Press:  01 March 2016

Simon Kapfer
Simon Kapfer*
Affiliation:
Laboratoire de Mathématiques et Applications, UMR CNRS 6086, Université de Poitiers, Téléport 2, Boulevard Marie et Pierre Curie, F-86962 Futuroscope Chasseneuil, France email simon.kapfer@math.univ-poitiers.fr

Abstract

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We study cup products in the integral cohomology of the Hilbert scheme of $n$ points on a K3 surface and present a computer program for this purpose. In particular, we deal with the question of which classes can be represented by products of lower degrees.

Supplementary materials are available with this article.

MSC classification

Primary: 14J10: Families, moduli, classification 14Q15: Higher-dimensional varieties
Secondary: 05A17: Partitions of integers 55N32: Orbifold cohomology
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 19 , Issue 1 , 2016 , pp. 78 - 97
Copyright
© The Author 2016 

References

Boissière, S., Nieper-Wißkirchen, M. and Sarti, A., ‘Smith theory and irreducible holomorphic symplectic manifolds’, J. Topol. 6 (2013) no. 2, 361–390.Google Scholar
Dénes, J., ‘The representation of a permutation as the product of a minimal number of transpositions, and its connection with the theory of graphs’, Publ. Math. Inst. Hung. Acad. Sci. 4 (1959) 63–71.Google Scholar
Ellingsrud, G., Göttsche, L. and Lehn, M., ‘On the Cobordism class of the Hilbert scheme of a surface’, J. Algebra. Geom. 10 (2001) 81–100.Google Scholar
Fogarty, J., ‘Algebraic families on an algebraic surface’, Amer. J. Math. 10 (1968) 511–521.CrossRefGoogle Scholar
Lascoux, A., Symmetric functions, Notes of the course given at Nankai University (2001),http://www.mat.univie.ac.at/∼slc/wpapers/s68vortrag/ALCoursSf2.pdf.Google Scholar
Lehn, M. and Sorger, C., ‘The cup product of Hilbert schemes for K3 surfaces’, Invent. Math. 152 (2003) no. 2, 305–329.CrossRefGoogle Scholar
Markman, E., ‘Integral generators for the cohomology ring of moduli spaces of sheaves over Poisson surfaces’, Adv. Math. 208 (2007) no. 2, 622–646.Google Scholar
Markman, E., ‘Integral constraints on the monodromy group of the hyperKähler resolution of a symmetric product of a K3 surface’, Internat. J. Math. 21 (2010) no. 2, 169–223.Google Scholar
Milnor, J. and Husemöller, D., Symmetric bilinear forms , Ergebnisse der Mathematik und ihrer Grenzgebiete 73 (Springer, Berlin, 1973).Google Scholar
Nakajima, H., ‘Heisenberg algebra and Hilbert schemes of points on projective surfaces’, Ann. of Math. (2) 145 (1997) no. 2, 379–388.Google Scholar
Qin, Z. and Wang, W., ‘Integral operators and integral cohomology classes of Hilbert schemes’, Math. Ann. 331 (2005) no. 3, 669–692.Google Scholar
Verbitsky, M., ‘Cohomology of compact hyperkähler manifolds and its applications’, Geom. Funct. Anal. 6 (1996) no. 4, 601–611.Google Scholar
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