Spherical subcategories in algebraic geometry (Q2825993)

For a fixed algebraically closed field \(\mathbf{k}\), let \(\mathcal{D}\) be a Hom-finite, \(\mathbf{k}\)-linear triangulated category.NEWLINENEWLINELet \(F\) be a \(d\)-spherelike object with Serre dual.NEWLINENEWLINEThe main theorems (Theorems 4.4, 4.6) of the paper under review give construction of a maximal triangulated subcategory containing \(F\) as a \(d\)-spherical object.NEWLINENEWLINEAn algebro-geometric application is the following (Proposition 5.5): if \(X\) is a Calabi-Yau variety (CY for short) and \(\pi:\widetilde{X}\to X\) denotes a successive blow-ups along points, the structure sheaf \(\mathcal{O}_{\widetilde{X}}\) is a spherelike object in the bounded derived category of coherent sheaves \(\mathcal{D}^{b}(\widetilde{X})\) and the maximal subcategory in the above theorem is \(\pi^{*}\mathcal{D}^{b}(X)\).NEWLINENEWLINEWe give the details of terminology.NEWLINENEWLINE\(F\) is \(d\)-\textit{spherelike} if \(\mathrm{Hom}^{\bullet}(F,F)=\mathbf{k}\oplus\mathbf{k}[-d]\).NEWLINENEWLINEIf is \(d\)-\textit{spherical} if it further satisfies \(d\)-\(CY\): \(\mathrm{Hom}^{\bullet}(F,A)\cong \mathrm{Hom}^{\bullet}(A,F[d])^{*}\) for any \(A\in \mathcal{D}\).NEWLINENEWLINEFor an object \(F\), a Serre dual of \(F\), denoted by \(\mathsf{S}F\), is defined as an object representing the functor \(\mathrm{Hom}(F,\cdot)^{*}\), i.e., \(\mathrm{Hom}(F,\cdot)^{*}\cong \mathrm{Hom}(\cdot,\mathsf{S}F)\).NEWLINENEWLINEIn particular if \(F\) is \(d\)-CY, \(\mathsf{S}F\cong F[d]\).NEWLINENEWLINEWe sketch the construction of the above maximal subcategory.NEWLINENEWLINEFor a \(d\)-spherelike object \(F\) in \(\mathcal{D}\), let \(\omega(F):=\mathsf{S}{F}[-d]\).NEWLINENEWLINESince \(\mathrm{Hom}(F,\omega(F))=\mathrm{Hom}(F,F)^{*}[-d]=\mathbf{k}\) unless \(d\neq0\), we have a twist NEWLINE\[NEWLINE F=\mathrm{Hom}(F,\omega(F))\otimes F \to \omega(F)\to Q_{F}. NEWLINE\]NEWLINENEWLINENEWLINEWe also have a similar twist even when \(d=0\) (Appendix).NEWLINENEWLINEThe above maximal subcategory is set to be the left orthogonal complement \(\mathcal{D}_{F}:={}^{\perp}Q_{F}\).NEWLINENEWLINEThe proof of the theorem consists of two parts: \(F\in \mathcal{D}_{F}\) (Lemma 4.1) and the maximality (Theorem 4.6).NEWLINENEWLINEThe latter fact is proved by checking \(\mathrm{Hom}^{\bullet}(\mathcal{U},Q_{F})=0\) using \(d\)-sphericity for any triangulated subcategory \(\mathcal{U}\) containing \(F\) as a \(d\)-spherical object.NEWLINENEWLINEFor the above mentioned application to CY of dimension \(d\), we use the well-known semi-orthogonal decomposition NEWLINE\[NEWLINE \mathcal{D}^{b}(\widetilde{X})=\langle \mathcal{O}_{E}(-(d-1)),\mathcal{O}_{E}(-(d-2)),\dots,\mathcal{O}_{E}(-1),\pi^{*} \mathcal{D}^{b}(X)\rangle, NEWLINE\]NEWLINE where \(\widetilde{X}\to X\) is assumed to be one-point blow-up and \(E\cong\mathbb{P}^{d-1}\) is the exceptional divisor.NEWLINENEWLINELet \(F:=\pi^{*}\mathcal{O}_{X}\) (in fact any pull-back of spherical object in \(\mathcal{D}^{b}(X)\) with support containing the blow-up center).NEWLINENEWLINEProposition 5.2 assures that \(\mathcal{D}_{F}=\pi^{*}\mathcal{D}^{b}(X)\) and its right orthogonal complement \(\mathcal{D}_{F}^{\perp}=\langle \mathcal{O}_{E}(-(d-1)),\mathcal{O}_{E}(-(d-2)),\dots,\mathcal{O}_{E}(-1)\rangle\).