Equivalence of symplectic singularities

Details Equivalence of symplectic singularities Equivalence of symplectic singularities

2011-02-04

arxiv.org/abs/1102.0865v5

Mathematics

Algebraic Geometry

26 pages, completely revised

Scientific paper

Let X be an affine normal variety with a C^*-action having only positive weights. Assume that X_{reg} has a symplectic 2-form w of weight l. We prove that, when l is not zero, the w is a unique symplectic 2-form of weight l up to C^*-equivariant automorphism When $l = 0$, we have a counter-example to this statement. In the latter half of the article, we associate to X a projective variety P(X) and prove that P(X) has a contact orbifold structure. Moreover, when X has canonical singularities, the contact orbifold structure is rigid under a small deformation. By using the contact structure on P(X), we discuss the equivalence problem for (X, w) up to contant. In most examples the symplectic structures turn out to be unique up to constant with very few exceptions.

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