Equivalence of symplectic singularities

TL;DR

This paper investigates the uniqueness of symplectic forms on affine varieties with C^*-actions, establishing conditions under which these forms are unique up to automorphism and exploring associated contact structures.

Contribution

It proves the uniqueness of symplectic 2-forms of non-zero weight up to automorphism and introduces a contact orbifold structure on an associated projective variety.

Findings

Symplectic forms are unique up to automorphism when weight l is not zero.

Counter-example exists when weight l is zero.

Associated projective variety has a rigid contact orbifold structure.

Abstract

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.