Fourier-Mukai functors: a survey (Q2869237)

Let \(X_1\) and \(X_2\) be two smooth projective varieties over a field and let \(\mathcal{E}\) be an object in \(D^b(X_1\times X_2)\), the bounded derived category of coherent sheaves on \(X_1\times X_2\). A Fourier--Mukai (FM) functor is an exact functor \(D^b(X_1)\to D^b(X_2)\) of the form \(F\mapsto (p_{X_2})_*(\mathcal{E}\otimes p_{X_1}^*F)\). The object \(\mathcal{E}\) is called the kernel. The article under review surveys many results in the theory of these functors.NEWLINENEWLINEIn the introduction the authors present some historical background and recall, in particular, Orlov's theorem stating that any fully faithful functor \(F: D^b(X_1)\to D^b(X_2)\) with a left adjoint is an FM-functor whose kernel is uniquely determined, and some of the generalisations of this statement. The story continues in Section 2 with a recollection of some standard properties of FM-functors. For instance, any FM-functor has a left and a right adjoint, induces functors on the Grothendieck groups, Hochschild homology and singular cohomology, which are, in a sense made precise in the paper, compatible with each other. The authors also survey the appearance of FM-functors in the study of bounded derived categories of \(K3\) surfaces.NEWLINENEWLINEIn Section 3 the main questions in the area are discussed. The construction described above defines a functor \(\Phi^{X_1\to X_2}_-: D^b(X_1\times X_2)\to \text{ExFun}(D^b(X_1),D^b(X_2))\), where the latter is the category of exact functors from \(D^b(X_1)\) to \(D^b(X_2)\), and it is natural to ask whether this functor is (a) essentially surjective, (b) essentially injective, (c) full or (d) faithful. One can also ask whether (e) \(\text{ExFun}(D^b(X_1),D^b(X_2))\) admits a triangulated structure such that \(\Phi^{X_1\to X_2}_-\) is exact. In particular, (a) asks for an extension of Orlov's theorem to non fully faithful functors. The authors also discuss the importance of the projectivity assumption. Furthermore, they, in particular, present results which show that the existence of an adjoint in Orlov's theorem is automatic and that any full functor is actually automatically faithful.NEWLINENEWLINEThe following section presents examples which show that the answers to (c), (d) and (e) are negative in general. It is also shown that FM-kernels are in general not unique, but their cohomology sheaves are uniquely determined.NEWLINENEWLINESection 5 is devoted to the extension of Orlov's result, that is, to question (a). The authors discuss several results available in the literature and some of the techniques needed to prove these, for instance, ample sequences, convolutions and differential graded categories. A particular generalisation which is presented is the supported case, that is, the situation where one works with a quasi-projective scheme \(X\) containing a projective scheme \(Z\) satisfying some conditions and where we consider derived categories supported on \(Z\).NEWLINENEWLINEThe paper concludes with a section where several open problems are discussed.NEWLINENEWLINEFor the entire collection see [Zbl 1256.14001].