Equivalence of symplectic singularities

Understand

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.