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 . Since the known examples of irreducible symplectic varieties have or , our variety is in a new deformation class. The example is obtained as follows. Let 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 with and . Let be this symplectic desingularization: there is a natural locally trivial fibration . Our example is the fiber over of this map, we denote it by . The main body of the paper is devoted to the proof that is irreducible symplectic and that . Applying the generalized Lefschetz Hyperplane Theorem we get that low-dimensional homotopy (or homology) groups of are represented by homotopy (or homology) groups of a subset of 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

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

Received by editor(s):
November 9, 2000

Published electronically:
January 14, 2003

Additional Notes:
Supported by Cofinanziamento MURST 1999-2001

Dedicated:
Dedicato a Riccardino