Equivalence of symplectic singularities

Abstract

After introducing an equivalence problem for symplectic singularities, we formulate an algebraic version of such a problem. Let $X$ be an affine normal variety with a ${\mathbf{C}}^{\ast }$-action having only positive weights. Assume that the regular part $X_{reg}$ of $X$ admits an algebraic symplectic $2$-form $\omega$ with weight $l$. Our main theorem asserts that any algebraic symplectic $2$-form $\omega \text{'}$ on $X_{reg}$ of weight $l$ is equivalent to $\omega$ up to a ${\mathbf{C}}^{\ast }$-equivariant automorphism of $X$ if $l\ne 0$. When $l=0$ we have a counterexample to this statement. In the latter half of the article, we discuss the equivalence problem up to constant. We associate to $X$ a projective variety $\mathbf{P}\left(X\right)$ and prove that $\mathbf{P}\left(X\right)$ has a contact orbifold structure. Moreover, when $X$ has canonical singularities, the contact orbifold structure is rigid under a small deformation. The equivalence problem is then reduced to the uniqueness of the contact structures. In most examples the symplectic structures turn out to be unique up to constant with very few exceptions. In the final section we pose a splitting conjecture for symplectic singularities.