Equivalence of symplectic singularities (Q2376824)

Let \(X\) be a germ of a normal complex space, and let \(\omega\) be a holomorphic symplectic 2-form on the regular locus \(X_{\text{reg}}\) of \(X\). Two such pairs \((X, \omega)\) and \((X', \omega')\) are called equivalent if there is an isomorphism \(\phi: X\to X'\), such that \(\omega=\phi^*(\omega')\). First, the author proves a Darboux theorem for quotient singularities. More precisely, if \(X\) is the germ of a quotient singularity, then for any two holomorphic symplectic forms \(\omega\) and \(\omega'\) on \(X_{\text{reg}}\), \((X, \omega)\) and \((X, \omega')\) are equivalent. In the rest of the paper, the author considers algebraic version of the equivalence problem. Let \((X, \omega)\) be a pair of an even dimensional normal affine variety with a \(\mathbb{C}^*\) action and an algebraic symplectic 2-form \(\omega\) on \(X_{\text{reg}}\). Assume the \(\mathbb{C}^*\) action has positive weights. If the weight of \(\omega\) is \(l\neq 0\), then \(\omega\) is the unique holomorphic symplectic form of weight \(l\), up to equivalence. A counter example in the case \(l=0\) is provided. The next main result is, under the above assumption the projective cone \(\mathbb{P}(X)\) has a contact orbifold structure. Moreover, when \(X\) has canonical singularities, the contact orbifold structure is rigid under a small flat deformation. Some immediate applications are the uniqueness of symplectic 2-forms on some hypersurfaces up to a constant. The paper ends with a few open questions, including a splitting conjecture of affine symplectic varieties, which is an analogue of Bogomolov splitting theorem.