Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



A new six-dimensional irreducible symplectic variety

Author: Kieran G. O'Grady
Journal: J. Algebraic Geom. 12 (2003), 435-505
Published electronically: January 14, 2003
MathSciNet review: 1966024
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Abstract | References | Additional Information

Abstract: We construct a six-dimensional irreducible symplectic variety with $b_2=8$. Since the known examples of irreducible symplectic varieties have $b_2=7$ or $b_2=23$, our variety is in a new deformation class. The example is obtained as follows. Let $J$ be the Jacobian of a genus-two curve with its natural principal polarization: results of another paper of ours give a symplectic desingularization of the moduli space of semistable rank-two sheaves on $J$ with $c_1=0$ and $c_2=2$. Let $\mathcal{M}_{\mathbf{v}}$ be this symplectic desingularization: there is a natural locally trivial fibration $\mathcal{M}_{\mathbf{v}}\rightarrow J\times\widehat{J}$. Our example is the fiber over $(0,\widehat{0})$ of this map, we denote it by $\widetilde{\mathcal{M}}$. The main body of the paper is devoted to the proof that $\widetilde{\mathcal{M}}$is irreducible symplectic and that $b_2(\widetilde{\mathcal{M}})=8$. Applying the generalized Lefschetz Hyperplane Theorem we get that low-dimensional homotopy (or homology) groups of $\widetilde{\mathcal{M}}$are represented by homotopy (or homology) groups of a subset of $\widetilde{\mathcal{M}}$ which has an explicit description. The main problem is to provide the explicit description and to extract the necessary information on homotopy or homology groups.

References [Enhancements On Off] (What's this?)

  • [B] A. Beauville, Variétés Kählériennes dont la première classe de Chern est nulle, J. Diff. Geom 18 (1983), 755-782.
  • [BDL] J. Bryan - R. Donagi - Naichung Conan Leung, $G$-Bundles on abelian surfaces, hyperkähler manifolds, and stringy Hodge numbers, AG/0004159 (2000).
  • [HG] D. Huybrechts - L. Göttsche, Hodge numbers of moduli spaces of stable bundles on $K3$ surfaces, Internat. J. of Math. 7 (1996), 359-372.
  • [G] D. Gieseker, On the moduli of vector bundles on an algebraic surface, Ann. of Math 106 (1977), 45-60.
  • [GM] M. Goresky - R. MacPherson, Stratified Morse Theory, Ergeb. Math. Grenzgeb. (3. Folge) 14, Springer, 1988.
  • [L1] J. Li, The first two Betti numbers of the moduli space of vector bundles on surfaces, Comm. Anal. Geom. 5 (1997), 625-684.
  • [L2] J. Li, Algebraic geometric interpretation of Donaldson's polynomial invariants, J. Differential Geometry 37 (1993), 417-466.
  • [Mor] J. Morgan, Comparison of the Donaldson polynomial invariants with their algebro geometric analogues, Topology 32 (1993), 449-488.
  • [Muk1] S. Mukai, Duality between $D(X)$ and $D(\widehat{X})$ with its application to Picard sheaves, Nagoya Math. J. 81 (1981), 153-175.
  • [Muk2] S. Mukai, Symplectic structure of the moduli space of sheaves on an abelian or $K3$ surface, Invent. Math 77 (1984), 101-116.
  • [Muk3] S. Mukai, On the moduli space of bundles on $K3$ surfaces, Vector bundles on Algebraic Varieties, Tata Institute Studies in Mathematics, Oxford University Press, 1987.
  • [Mum] D. Mumford, Abelian Varieties, Tata Institute Studies in Mathematics, Oxford University Press, 1970.
  • [O1] K. O'Grady, Desingularized moduli spaces of sheaves on a $K3$, J. Reine Angew. Math. 512 (1999), 49-117.
  • [O2] K. O'Grady, The weight-two Hodge structure of moduli spaces of sheaves on a $K3$ surface, J. Algebraic Geom. 6 (1997), 599-644.
  • [O3] K. O'Grady, Moduli of vector-bundles on surfaces, Algebraic Geometry Santa Cruz 1995, Proc. Symp. Pure Math. vol. 62, Amer. Math. Soc., 1997, pp. 101-126.
  • [O4] K. O'Grady, Relations among Donaldson polynomials of certain algebraic surfaces, I, Forum Math. 8 (1996), 1-61.
  • [Y1] K. Yoshioka, Some examples of Mukai's reflections on $K3$ surfaces, J. Reine Angew. Math. 515 (1999), 97-123.
  • [Y2] K. Yoshioka, Moduli spaces of stable sheaves on abelian surfaces, AG/0009001 (2000).
  • [W] F. W. Warner, Foundations of Differentiable Manifolds and Lie Groups, Graduate Texts in Mathematics 94, Springer-Verlag, 1983.

Additional Information

Kieran G. O'Grady
Affiliation: Università La Sapienza, Dipartimento di Matematica G. Castelnuovo, Piazzale A Moro 5, 00185 Rome, Italy

Received by editor(s): November 9, 2000
Published electronically: January 14, 2003
Additional Notes: Supported by Cofinanziamento MURST 1999-2001
Dedicated: Dedicato a Riccardino

American Mathematical Society