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Singular symplectic moduli spaces. (English) Zbl 1096.14037

Apart from those constructed by S. Mukai [Invent. Math. 77, 101–116 (1984; Zbl 0565.14002)] there is only one known example of an irreducible holomorphic symplectic manifold. Mukai’s examples are the moduli spaces \(M_v\) of semistable sheaves on a polarised \(K3\) or abelian surface in the cases where no strictly semistable sheaves exist. When there do exist strictly semistable sheaves, \(M_v\) is singular and one may ask whether \(M_v\) admits a projective symplectic resolution. This question has been answered in two cases by O’Grady, yielding two new deformation classes of irreducible holomorphic symplectic manifolds.
In this paper, the authors show that, apart from O’Grady’s examples and examples arising from symmetric products of \(K3\) or abelian surfaces, no further examples arise in this way. The key to this is proving that the moduli spaces \(M_v\) are locally factorial and, apart from the special cases mentioned, have singularities in codimension at least \(4\).

MSC:

14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
14D20 Algebraic moduli problems, moduli of vector bundles
14J28 \(K3\) surfaces and Enriques surfaces
32J27 Compact Kähler manifolds: generalizations, classification
53D30 Symplectic structures of moduli spaces

Citations:

Zbl 0565.14002
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References:

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