Title: Symplectic singularities of varieties: The method of algebraic restrictions
Abstract:We study germs of singular varieties in a symplectic space. In [V. I. Arnold, First step of local symplectic algebra, Differential topology, infinite-dimensional Lie algebras, and applications, D. B. ...We study germs of singular varieties in a symplectic space. In [V. I. Arnold, First step of local symplectic algebra, Differential topology, infinite-dimensional Lie algebras, and applications, D. B. Fuchs' 60th anniversary collection, American Mathematical Society Transl. Ser. 2, Am. Math. Soc. 194(44) (1999), 1–8.], V. Arnold discovered so called “ghost” symplectic invariants which are induced purely by singularity. We introduce algebraic restrictions of differential forms to singular varieties and show that this ghost is exactly the invariants of the algebraic restriction of the symplectic form. This follows from our generalization of Darboux-Givental' theorem from non-singular submanifolds to arbitrary quasi-homogeneous varieties in a symplectic space. Using algebraic restrictions we introduce new symplectic invariants and explain their geometric meaning. We prove that a quasi-homogeneous variety N is contained in a non-singular Lagrangian submanifold if and only if the algebraic restriction of the symplectic form to N vanishes. The method of algebraic restriction is a powerful tool for various classification problems in a symplectic space. We illustrate this by complete solutions of symplectic classification problem for the classical A, D, E singularities of curves, the S5 singularity, and for regular union singularities.Read More
Publication Year: 2008
Publication Date: 2008-01-01
Language: en
Type: article
Indexed In: ['crossref']
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Cited By Count: 17
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