Derived categories of singular surfaces. (English) Zbl 1493.14023

Karmazyn, Joseph; Kuznetsov, Alexander; Shinder, Evgeny

J. Eur. Math. Soc. (JEMS) 24, No. 2, 461-526 (2022).

This paper begins by introducing a fairly general approach to construct a semiorthogonal decomposition (SOD) of the bounded derived category of coherent sheaves \(D^b(X)\) of a normal projective surface \(X\) with rational singularities (over an algebraically closed field of characteristic 0). The main idea is to consider a resolution of singularities \(\pi: \widetilde{X}\rightarrow X\), and induce an SOD from an SOD of \(D^b(\widetilde{X})=\langle \widetilde{\mathcal{A}_1}, \dots, \widetilde{\mathcal{A}_n}\rangle\) compatible with the exceptional divisor components, where each connected exceptional divisor component \(D_i\) corresponds to an SOD component \(\mathcal{A}_i= \pi_*\widetilde{\mathcal{A}_i}=\widetilde{\mathcal{A}_i}/\langle \mathcal{O}_E(-1)\rangle_{E\subset D_i} \subset D^b(X)\). The authors further observe that if \(\pi\) is crepant along \(D_i\) for \(i>2\), the above SOD also induces SOD of the derived category of perfect complexes by simply intersecting with \(\mathcal{A}_i\), and these SOD components of perfect complexes are admissible when \(\pi\) is a crepant resolution.
The proof of the above results is an adaptation of the method developed by the second named author. The first observation is that SOD of \(D^b(\widetilde{X})\) extends to the bounded above category \(D^-(\widetilde{X})\), and \((\pi^*, \pi_*)\) is an adjoint pair on the bounded above category (but not on the bounded category or perfect complexes). This decomposes \(D^-(\widetilde{X})=\langle \ker \pi_*, \pi_*D^-(\widetilde{X}) \rangle\). The authors provide a technical argument that one can produce the same SOD components of \(D^b(X)\) in two ways: (1) by taking \(\pi_*\) of the SOD components of \(D^-(\widetilde{X})\) then restricted to \(D^b(X)\) (2) by first killing \(\ker \pi_*\cap \widetilde{\mathcal{A}_k}\) inside each SOD component \(\widetilde{\mathcal{A}_k}\) of \(D^b(\widetilde{X})\) (which is shown to be the smallest triangulated category closed under the direct sums of \(\mathcal{O}_{E_{i,k}}(-1)\), \(E_{i,k}\subset D_k\)) then apply \(\pi_*\). In the crepant case, the bounded SOD components are also preserved by \(\pi^*\), leading to nicer results on perfect complexes.
The authors proceed by showing that when \(X\) has cyclic quotient singularities, \(\mathcal{A}_i\) can be constructed more explicitly under another assumption that \(\widetilde{\mathcal{A}_k}\) (twisted) adheres to the chain of curves \(\cup_i E_{i,k}\), which means a full exceptional collection of line bundles \(\widetilde{\mathcal{A}_k}=\langle \mathcal{L}_0, \mathcal{L}_0(E_{1,k}), \dots, \mathcal{L}_0(\sum_i E_{i,k}) \rangle\). Morally, this makes \(\widetilde{\mathcal{A}_k}\) the smallest admissible subcategory containing \(\mathcal{O}_{E_{i,k}}(-1)\). The twist i.e.self-intersections of \(E_{i,k}\) that comes with the assumption identifies an element in the Brauer group \(Br(X)\). The SOD of the twisted derived category by the element is described using reformulation of a result of Hille and Ploog, where the SOD components are identified as derived categories of a finite dimensional \(k\)-algebras.
These results are further illustrated in the case of toric surfaces, and related to iterative extension of rank 1 reflexive sheaves considered by Y. Kawamata [Compos. Math. 154, No. 9, 1815–1842 (2018; Zbl 1423.14017)].

Reviewer: Jesse Huang (Edmonton)

MSC:

14F08	Derived categories of sheaves, dg categories, and related constructions in algebraic geometry
14M25	Toric varieties, Newton polyhedra, Okounkov bodies
14J17	Singularities of surfaces or higher-dimensional varieties
14F22	Brauer groups of schemes

Keywords:

derived categories; semiorthogonal decompositions; toric surfaces; Brauer group

Citations:

Zbl 1423.14017

References:

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