Derived equivalences by quantization (Q2427022)

Let \(X\) be a smooth scheme of finite type over a field \(k\), equipped with a projective birational map \(\pi : X\rightarrow Y\) onto a normal irreducible affine scheme of finite type over \(k\). The main result of the paper under review asserts that if \(\text{char}\, k = 0\) and \(X\) is symplectic (i.e., it admits a non-degenerate closed 2-form \(\Omega \in H^0(X,{\Omega}^2_X)\)) then every point \(y\in Y\) has an étale neighborhood \(U_y \rightarrow Y\) such that there exists a vector bundle \(\mathcal E\) on the pullback \(X_y = X{\times}_YU_y\) which is a tilting generator of the derived category \(D^b_{\text{coh}}(X)\). The last assertion means that: (i) \({\text{Ext}}^i({\mathcal E},{\mathcal E}) = 0\), \(\forall i > 0\), and (ii) \(\forall {\mathcal F}^{\bullet} \in \text{Ob}\, D^-_{\text{coh}}(X_y)\), \({\text{RHom}}^{\bullet}({\mathcal E},{\mathcal F}^{\bullet}) \simeq 0\) implies \({\mathcal F}^{\bullet} \simeq 0\). In this case, \({\mathcal F}^ {\bullet} \mapsto {\text{RHom}}^{\bullet}({\mathcal E},{\mathcal F}^{\bullet})\) is an equivalence of categories \(D^b_{\text{coh}}(X_y) \rightarrow D^b_{\text{coh}}(R\text{-mod})\), where \(R = \text{End}({\mathcal E})\) and \(R\)-mod is the category of finitely generated left \(R\)-modules (which is abelian since it turns out that \(R\) is left noetherian). The author also shows that if \({\pi}^{\prime} : X^{\prime} \rightarrow Y\) is another resolution of singularities and if the canonical bundles \(K_X\) and \(K_{X^{\prime}}\) are trivial, then every point \(y \in Y\) admits an étale neighborhood \(U_y \rightarrow Y\) such that \(D^b_{\text{coh}}(X_y)\) and \(D^b_{\text{coh}}(X^{\prime}_y)\) are equivalent. The author proves these results by reduction to positive characteristic. He uses his results from \textit{D. Kaledin} [Math. Res. Lett. 13, No. 1, 99--107 (2006; Zbl 1090.53064)] about the existence of twistor deformations of line bundles on \(X\), and a quantization result in positive characteristic based on the techniques developed in \textit{R. Bezrukavnikov} and \textit{D. Kaledin} [J. Am. Math. Soc. 21, No. 2, 409--438 (2008; Zbl 1138.53067)].