A new six-dimensional irreducible symplectic variety

Author:
Kieran G. O’Grady

Journal:
J. Algebraic Geom. **12** (2003), 435-505

DOI:
https://doi.org/10.1090/S1056-3911-03-00323-0

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.

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[O1]O1 K. O’Grady, *Desingularized moduli spaces of sheaves on a $K3$*, J. Reine Angew. Math. **512** (1999), 49-117.
[O2]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]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.
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[Y2]Y2 K. Yoshioka, *Moduli spaces of stable sheaves on abelian surfaces*, AG/0009001 (2000).
[W]W F. W. Warner, *Foundations of Differentiable Manifolds and Lie Groups*, Graduate Texts in Mathematics 94, Springer-Verlag, 1983.

[B]B A. Beauville, *Variétés Kählériennes dont la première classe de Chern est nulle*, [1] J. Diff. Geom **18** (1983), 755-782.
[BDL]BDL J. Bryan - R. Donagi - Naichung Conan Leung, *$G$-Bundles on abelian surfaces, hyperkähler manifolds, and stringy Hodge numbers*, AG/0004159 (2000).
[HG]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]G D. Gieseker, *On the moduli of vector bundles on an algebraic surface*, Ann. of Math **106** (1977), 45-60.
[GM]GM M. Goresky - R. MacPherson, *Stratified Morse Theory*, Ergeb. Math. Grenzgeb. [1] (3. Folge) 14, Springer, 1988.
[L1]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]L2 J. Li, *Algebraic geometric interpretation of Donaldson’s polynomial invariants*, [1] J. Differential Geometry **37** (1993), 417-466.
[Mor]Mor J. Morgan, *Comparison of the Donaldson polynomial invariants with their algebro geometric analogues*, Topology **32** (1993), 449-488.
[Muk1]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]Muk2 S. Mukai, *Symplectic structure of the moduli space of sheaves on an abelian or $K3$ surface*, Invent. Math **77** (1984), 101-116.
[Muk3]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]Mum D. Mumford, *Abelian Varieties*, Tata Institute Studies in Mathematics, Oxford University Press, 1970.
[O1]O1 K. O’Grady, *Desingularized moduli spaces of sheaves on a $K3$*, J. Reine Angew. Math. **512** (1999), 49-117.
[O2]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]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]O4 K. O’Grady, *Relations among Donaldson polynomials of certain algebraic surfaces, I*, Forum Math. **8** (1996), 1-61.
[Y1]Y1 K. Yoshioka, *Some examples of Mukai’s reflections on $K3$ surfaces*, J. Reine Angew. Math. **515** (1999), 97-123.
[Y2]Y2 K. Yoshioka, *Moduli spaces of stable sheaves on abelian surfaces*, AG/0009001 (2000).
[W]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

Email:
ogrady@mat.uniroma1.it

Received by editor(s):
November 9, 2000

Published electronically:
January 14, 2003

Additional Notes:
Supported by Cofinanziamento MURST 1999-2001

Dedicated:
Dedicato a Riccardino