Autoequivalences of derived categories on the minimal resolutions of \(A_n\)-singularities on surfaces
Let \(X\) be a smooth projective variety and \(D(X)\) the bounded derived category of \(X\). The derived category \(D(X)\) carries a lot of birational information on \(X\). One natural question is to understand the group of isomorphism classes of autoequivalences of \(D(X)\), denoted by \(\text{Auteq\,}D(X)\). Let \(A\) be the subgroup of \(\text{Auteq\,}D(X)\) generated by (1) automorphims of \(X\), (2) tensoring with line bundles and (3) shift functor. When \(K_X\) or \(-K_X\) is ample, Bondal and Orlov showed that \(A\) is the whole autoequivalence group. In general, one expects that \(\text{Auteq\,}D(X)\) contains more elements. In the paper under review, the authors study the crepant resolution for \(A_n\)-singularities. Let \(Y=\text{Spec} {\mathbb C}x,y,z/(x^2+y^2+z^{n+1})\) be the \(A_n\)-singularity, \(P \in Y\) the closed point. Consider the crepant resolution \(f: X \to Y\) and \(Z:=f^{-1}(P)\). The authors study the generators of \(\text{Auteq\,}D_Z(X)\), where \(D_Z(X)\) is the ``local derived category'', i.e. the full subcategory of objects supported on the fiber \(Z\). Since \(K_X\) is trivial on the the exceptional divisor \(Z\), one expects there are more elements in \(\text{Auteq\,}D_Z(X)\). Recall that \( \alpha \in D_{Z}(X)\) is spherical if (1) \(\alpha \otimes \omega_X \cong \alpha\), and (2) \( \text{Hom}(\alpha, \alpha)=0\) and \(\text{Hom}^k(\alpha, \alpha)={\mathbb C}\) when \(k \neq 0\). For any spherical object, one can associate a natural twist functor \(T_{\alpha}\). These twist functors play an important role in the study of homological mirror symmetry. Go back to the \(A_n\) case. The exceptional divisor \(Z\) is a chain of rational curves. There are a lot of spherical objects coming from sheaves supported on exceptional curves. Let \(B=\langle T_{\alpha} |\alpha \in D_Z(X), {\text{ spherical}}\rangle\). One is particularly interested in autoequivalences which are given as Fourier-Mukai functors. Consider the subgroup \(\text{Auteq}^{\text{FM}}D_Z(X) \subset \text{Auteq}\,D_Z(X)\) consisting of Fourier-Mukai transforms. (Note that not all autoequivalences of \(D_Z(X)\) are given by Fourier-Mukai transforms.) One of the main results is that the group \(\text{Auteq}^{\text{FM}}D_Z(X)\) is generated by \(B\), Pic\((X)\) and the shift functor. One of the most technical proposition is the following: Key Proposition. For any \(\Psi \in \text{Auteq\,}D_Z(X)\), there exists an integer \(i\) and \(\Phi \in B\) such that \(\Phi \circ \Psi\) sends every skyscraper sheaf \({\mathcal O}_x\) with \(x \in Z\) to \({\mathcal O}_y[i]\) for some \(y \in Z\). The main results follow from this proposition and induction.
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