Extension of 2-forms and symplectic varieties (Q2744711)

The author proves two theorems (Stability theorem, Local Torelli theorem) for symplectic varieties. Let \(X\) be a good representative of a normal singularity. Then the singularity is symplectic if the regular locus \(U\) of \(X\) admits an everywhere non-degenerate holomorphic closed 2-form \(\omega\) where \(\omega\) extends to a regular form on \(Y\) for a resolution of singularities \(Y\rightarrow X.\) Similarly say that a normal compact Kähler space \(Z\) is a symplectic variety if the regular locus \(V\) of \(Z\) admits a non-degenerate holomorphic closed 2-form \(\omega\) where \(\omega\) extends to a regular form on \(\widetilde{Z},\) where \(\widetilde{Z}\rightarrow Z\) is a resolution of singularieties of \(Z.\) When \(Z\) has a resolution \(\pi:\widetilde{Z}\rightarrow Z\) such that \((\widetilde{Z}, \pi^{*}\omega)\) is a symplectic manifold, it is said \(Z\) has a symplectic resolution. NEWLINENEWLINENEWLINETheorem 7 (Stability theorem): Let \((Z, \omega)\) be a projective symplectic variety. Let \(g: L\rightarrow \Delta\) be a projective flat morphism from \(L\) to a 1-dimensional unit disc \(\Delta\) with \(g^{-1}(0) = Z.\) Then \(\omega\) extends sideways in the flat family so that it gives a symplectic 2-form \(\omega_t\) on each fiber \(Z_t\) for \(t\in \Delta_{\varepsilon}\) with a sufficiently small \(\varepsilon.\)