Fourier-Mukai functors in the supported case (Q2921070)

\textit{D. O. Orlov} [J. Math. Sci., New York 84, No. 5, 1361--1381 (1997; Zbl 0938.14019)] proved that for smooth projective varieties \(X_1\) and \(X_2\) over a field \(k\) any full and faithful exact functor \(\mathsf{F} \colon {D}^b(X_1)\to {D}^b(X_2)\) can be represented by an integral transform functor with kernel \(\mathcal{E} \in {D}^b(X_1 \times X_2)\), that is, \(\mathsf{F}(-) \cong \mathsf{R}p_{2\,*}(\mathcal{E} \otimes (p_1)^*(-))\), where \(p_i\) denotes the projection onto \(X_i\) for \(i \in \{1, 2\}\).NEWLINENEWLINEIn the literature there have been several variants, in particular the category \({D}^b(X)\) is replaced by \(\mathbf{Perf}(X)\), its subcategory of perfect complexes. This is the approach taken in [\textit{V. A. Lunts} and \textit{D. O. Orlov}, J. Am. Math. Soc. 23, No. 3, 853--908 (2010; Zbl 1197.14014)]. It was shown that a fully faithful functor between the corresponding categories of perfect complexes is represented by an integral transform, thus generalizing the previous result to possibly singular schemes. The main ingredient in their proof is the existence of enhancements, i.e., nice DG-models of the derived categories.NEWLINENEWLINEIn this paper, the authors keep pursuing Orlov's question about whether all exact functors between the categories of perfect complexes on projective schemes are of Fourier--Mukai type, focusing in the supported situation. Specifically, let \(Z \subset X\) be a closed subset and consider the full subcategory \(\mathbf{Perf}_Z(X)\) of \(\mathbf{Perf}(X)\) formed by complexes supported at \(Z\). The main result is the following. Let \(X_1\) be a quasi-projective scheme over a field \(k\) and \(Z_1\) be a projective subscheme such that all of its infinitesimal embeddings have a structure sheaf that is perfect as an object in \({D}(\mathbf{Qco}(X_1))\). Let \(X_2\) be a separated scheme of finite type over \(k\) and let \(Z_2\) be a closed subscheme of \(X_2\). An exact functor NEWLINE\[NEWLINE \mathsf{F} \colon \mathbf{Perf}_{Z_1}(X_1) \longrightarrow \mathbf{Perf}_{Z_2}(X_2) NEWLINE\]NEWLINE satisfying a certain technical condition \((*)\) is an integral functor such that \(\mathsf{F}(-) \cong \mathsf{R}p_{2\,*}(\mathcal{E} \otimes (p_1)^*(-))\) with a kernel \(\mathcal{E} \in {D}_{Z_1 \times Z_2}(\mathbf{Qco}(X_1 \times X_2))\). The condition \((*)\) is too long to repeat here. Suffice it to say that it is satisfied whenever \(\mathsf{F}\) is full and the 0-th dimensional torsion of \(\mathcal{O}_{Z_1}\) vanishes. The authors show that the technical condition \((*)\) is of potential application for certain non-full exact functors.NEWLINENEWLINEThe proof follows the basic ideas in Lunts--Orlov, with the needed changes to deal with the supported situation. In particular they prove that a quasi-projective scheme \(X\) over a field and a projective subscheme \(Z\) such that the 0-th dimensional torsion of \(\mathcal{O}_{Z}\) vanishes and all of its infinitesimal embeddings have a perfect structure, then \(\mathbf{Perf}_Z(X)\) has a strongly unique enhancement.NEWLINENEWLINEThe authors show that their results can be applied to 2-Calabi-Yau categories arising from local resolutions of \(A_n\)-singularities and also to 3-Calabi-Yau categories defined by the the total space of the canonical bundle of \(\mathbb{P}^2 \) supported on its zero section.