Poisson deformations and birational geometry (Q2788594)

Let \(X =\mathrm{Spec} R\) be an affine symplectic variety, where \(R\) is a graded ring \(R=\bigoplus_{i \geq 0}R_i\) with \(R_0 = \mathbb{C}\). Assume that the symplectic form on \(X\) is homogeneous with respect to the natural \(\mathbb{C}^*\)-action and that there exists a projective crepant resolution \(\pi: Y \to X\). The cohomology group \(H := H^2(Y, \mathbb{C})\) can be interpreted as the base space of the universal Poisson deformation \(\mathcal{Y} = (\mathcal{Y}_t)_{t\in H}\) of \(Y\), where \(\mathcal{Y}_0 \cong Y\). The base space \(Q\) of the universal Poisson deformation \(\mathcal{X} = (\mathcal{X}_s)_{s\in Q}\) of \(X\) then arises as \(Q = H/W\), where \(W\) is a finite group. There is a natural map \(\mathcal{Y} \to \mathcal{X}\), whose restriction \(\mathcal{Y}_0 \to \mathcal{X}_0\) coincides with \(\pi\). This map induces a map \(\Pi: \mathcal{Y} \to \mathcal{X}'\) over \(H\), where \(\mathcal{X}'\) is the pullback of the family \(\mathcal{X}\) to \(H\). The restrictions \(\Pi_t: \mathcal{Y}_t \to \mathcal{X}'_t\) are birational for all \(t\in H\) and isomorphisms for general \(t\in H\).NEWLINENEWLINEThe article under review studies the locus \(\mathcal{D} \subset H\) consisting of those \(t\in H\) where \(\Pi_t\) is not an isomorphism. As the first part of his main theorem, the author proves that there exists a finite number of hyperplanes \(H_i \subset H^2(Y, \mathbb{Q})\) such that \(\mathcal{D} = \bigcup_i H_i \otimes \mathbb{C}\). The second part of the main theorem asserts that there are only finitely many crepant projective resolutions \(\pi_k: Y_k \to X\) of \(X\). Their (relative) ample cones \(\mathrm{Amp}(\pi_k)\) can be viewed as open subsets of \(H^2(Y, \mathbb{R}) \cong \mathrm{Pic}(Y) \otimes \mathbb{R}\). It is the final part of the main theorem that the set of open chambers in \(H^2(Y, \mathbb{R})\) determined by the real hyperplanes \(H_i \otimes \mathbb{R}\) coincides with the set \(\{w(\mathrm{Amp}(\pi_k))\mid w \in W, \text{}\pi_k\) crepant projective resolution of \(X\)