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
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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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