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Symplectic singularities. (English) Zbl 0958.14001

The author introduces symplectic singularities and classifies symplectic singularities of the simplest type. A normal variety is said to have symplectic singularities if its smooth part carries a closed symplectic form, whose pull back to any resolution extends to a global holomorphic 2-form. The relation between symplectic singularities and symplectic complex (hyperKähler) manifolds is analogous to that between rational Gorenstein singularities and Calabi-Yau manifolds.
The main theorem states that a germ of an isolated singularity with a smooth projective tangent cone is symplectic if and only if it is analytically isomorphic to the germ \((\overline{\mathcal O}_{\text{min}}, 0)\), where \(\overline{\mathcal O}_{\text{min}}= {\mathcal O}_{\text{min}}\cup\{0\}\) is the closure of a smallest non-zero nilpotent orbit \({\mathcal O}_{\text{min}}\) for the adjoint action of a simple complex Lie algebra. As the next step in the classification of symplectic singularities the author proposes isolated singularities with trivial local fundamental group.

MSC:

14B05 Singularities in algebraic geometry
32S45 Modifications; resolution of singularities (complex-analytic aspects)
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
53C26 Hyper-Kähler and quaternionic Kähler geometry, “special” geometry
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