Beauville, Arnaud Symplectic singularities. (English) Zbl 0958.14001 Invent. Math. 139, No. 3, 541-549 (2000). 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. Reviewer: Eugenii I.Shustin (Tel Aviv) Cited in 5 ReviewsCited in 107 Documents 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 Keywords:symplectic singularities; normal variety; rational Gorenstein singularities; Calabi-Yau manifolds PDFBibTeX XMLCite \textit{A. Beauville}, Invent. Math. 139, No. 3, 541--549 (2000; Zbl 0958.14001) Full Text: DOI arXiv