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The standard conjectures for holomorphic symplectic varieties deformation equivalent to Hilbert schemes of K3 surfaces

Part of: Cycles and subschemes Global differential geometry Surfaces and higher-dimensional varieties

Published online by Cambridge University Press:  07 February 2013

François Charles  and
Eyal Markman
François Charles
Affiliation:
Département de Mathématiques et Applications, École Normale Supérieure, 45, rue d’Ulm, 75005 Paris, France (email: [email protected], [email protected])
Eyal Markman
Affiliation:
Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003, USA (email: [email protected])
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Abstract

We prove the standard conjectures for complex projective varieties that are deformations of the Hilbert scheme of points on a K3 surface. The proof involves Verbitsky’s theory of hyperholomorphic sheaves and a study of the cohomology algebra of Hilbert schemes of K3 surfaces.

Keywords

algebraic cyclesholomorphic symplectic varietiesstandard conjectures

MSC classification

Secondary: 14C25: Algebraic cycles 14J60: Vector bundles on surfaces and higher-dimensional varieties, and their moduli 53C26: Hyper-Kähler and quaternionic Kähler geometry, “special” geometry
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 3 , March 2013 , pp. 481 - 494
Copyright
Copyright © 2013 The Author(s)

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References

[And96]André, Y., Pour une théorie inconditionnelle des motifs, Pub. Math. Inst. Hautes Études Sci. 83 (1996), 5–49.CrossRefGoogle Scholar
[Ara06]Arapura, D., Motivation for Hodge cycles, Adv. Math. 207 (2006), 762–781.CrossRefGoogle Scholar
[Bea83]Beauville, A., Variétés Kähleriennes dont la première classe de Chern est nulle, J. Differential Geom. 18 (1983), 755–782.CrossRefGoogle Scholar
[Cal00]Căldăraru, A., Derived categories of twisted sheaves on Calabi–Yau manifolds, Thesis, Cornell University (May 2000).Google Scholar
[Cha10]Charles, F., Remarks on the Lefschetz standard conjecture and hyperkähler varieties, Comm. Math. Helvetici., to appear, Preprint (2010), arXiv:1002.5011v3.Google Scholar
[ES93]Ellingsrud, G. and Strømme, S. A., Towards the Chow ring of the Hilbert scheme of ℙ2, J. Reine Angew. Math. 441 (1993), 33–44.Google Scholar
[Got94]Göttsche, L., Hilbert schemes of zero-dimensional subschemes of smooth varieties, Lecture Notes in Mathematics, vol. 1572 (Springer, Berlin, 1994).CrossRefGoogle Scholar
[Gro69]Grothendieck, A., Standard conjectures on algebraic cycles, in Algebraic Geometry (Internat. Colloq., Tata Institute Fundamental Research, Bombay, 1968) (Oxford University Press, London, 1969), 193–199.Google Scholar
[HL97]Huybrechts, D. and Lehn, M., The geometry of moduli spaces of sheaves, Aspects of Mathematics, vol. E31 (Friedr. Vieweg & Sohn, Braunschweig, 1997).CrossRefGoogle Scholar
[Huy99]Huybrechts, D., Compact Hyperkähler manifolds: basic results, Invent. Math. 135 (1999), 63–113; and Erratum: Invent. Math. 152 (2003), 209–212.CrossRefGoogle Scholar
[Huy06]Huybrechts, D., Fourier–Mukai transforms in algebraic geometry (Oxford University Press, London, 2006).CrossRefGoogle Scholar
[Kle68]Kleiman, S., Algebraic cycles and the Weil conjectures, in Dix exposés sur la cohomologie des schémas (North-Holland, Amsterdam, 1968), 359–386.Google Scholar
[Kle94]Kleiman, S., The standard conjectures, Proceedings of Symposia in Pure Mathematics, vol. 55, part 1 (American Mathematical Society, Providence, RI, 1994), 3–20.Google Scholar
[LQW02]Li, W.-P., Qin, Z. and Wang, W., Vertex algebras and the cohomology ring structure of Hilbert schemes of points on surfaces, Math. Ann. 324 (2002), 105–133.CrossRefGoogle Scholar
[Lie68]Lieberman, D. I., Numerical and homological equivalence of algebraic cycles on Hodge manifolds, Amer. J. Math. 90 (1968), 366–374.CrossRefGoogle Scholar
[Mar]Markman, E., The Beauville–Bogomolov class as a characteristic class, Electronic preprint, arXiv:1105.3223.Google Scholar
[Mar02]Markman, E., Generators of the cohomology ring of moduli spaces of sheaves on symplectic surfaces, J. Reine Angew. Math. 544 (2002), 61–82.Google Scholar
[Mar07]Markman, E., Integral generators for the cohomology ring of moduli spaces of sheaves over Poisson surfaces, Adv. Math. 208 (2007), 622–646.CrossRefGoogle Scholar
[Mar08]Markman, E., On the monodromy of moduli spaces of sheaves on K3 surfaces, J. Algebraic Geom. 17 (2008), 29–99.CrossRefGoogle Scholar
[Muk87]Mukai, S., On the moduli space of bundles on K3 surfaces I, in Vector bundles on algebraic varieties, Proc. Bombay Conference, 1984, Tata Institute of Fundamental Research Studies, vol. 11 (Oxford University Press, London, 1987), 341–413.Google Scholar
[Muk81]Mukai, S., Duality between D(X) and $D(\hat X)$ with its application to Picard sheaves, Nagoya Math. J. 81 (1981), 153–175.CrossRefGoogle Scholar
[Ver99]Verbitsky, M., Hyperholomorphic sheaves and new examples of hyperkähler manifolds, alg-geom/9712012, in Hyperkähler manifolds, Mathematical Physics, vol. 12, eds Kaledin, D. and Verbitsky, M. (International Press, Somerville, MA, 1999).Google Scholar
[Yos01]Yoshioka, K., Moduli spaces of stable sheaves on abelian surfaces, Math. Ann. 321 (2001), 817–884.CrossRefGoogle Scholar