Cautis, Sabin; Kamnitzer, Joel; Licata, Anthony Derived equivalences for cotangent bundles of Grassmannians via categorical \(\mathfrak{sl}_2\) actions. (English) Zbl 1282.14034 J. Reine Angew. Math. 675, 53-99 (2013). A categorical \(\mathfrak{sl}_2\) action consists of a sequence of linear additive categories \(\mathcal D(-N),\ldots,\mathcal D(N)\) and, for each \(\lambda\in \mathbb Z\), a collection of functors \(E(\lambda):\mathcal D(\lambda -1)\rightarrow \mathcal D(\lambda +1)\) and \(F(\lambda):\mathcal D(\lambda +1)\rightarrow \mathcal D(\lambda -1)\) satisfying certain relations. These functors induce maps of vector spaces on the complexified (split) Grothendieck groups \(V(\lambda):=K(\mathcal D(\lambda))\otimes_{\mathbb Z}\mathbb C\) and the relations above induce an \(\mathrm{SL}_2\) action on \(V=\bigoplus V(\lambda)\). The reflection element then induces an isomorphism of vector spaces \(V(\lambda)\rightarrow V(-\lambda)\). Thus, in this context, a natural question is whether this isomorphism lifts to an equivalence of categories \(T:\mathcal D(\lambda)\rightarrow \mathcal (-\lambda)\).The authors answer this question in the case where \(\mathcal D(\lambda)\) are triangulated categories. This is a modified version of the Chuang-Rouquier construction [J. Chuang and R. Rouquier, Ann. Math. (2) 167, No. 1, 245–298 (2008; Zbl 1144.20001)], who deal with the case when the weight space categories \(\mathcal D(\lambda)\) are abelian and the categorical \(\mathfrak{sl}_2\) action is by exact functors.For their setup, the authors define a strong categorical \(\mathfrak{sl}_2\) action, which essentially means adding a grading to the \(\mathcal D(\lambda)\)’s and extra functors \(E^{(r)}(\lambda)\) and \(F^{(r)}(\lambda)\). Adapting the methods in [Zbl 1144.20001], the authors obtain the desired equivalence between \(\mathcal D(-\lambda)\) and \(\mathcal D(\lambda)\).As an application, they construct equivalences between derived categories of coherent sheaves on cotangent bundles of Grassmannians. More precisely, they obtain an equivalence \(\text{DCoh}(T^*(\mathrm{Gr}(k,N))\cong\text{DCoh}(T^*(\mathrm{Gr}(N-k,N))\). More precisely, the authors show that this equivalence is induced by a Fourier-Mukai kernel which is a Cohen-Macaulay sheaf. When \(k=1\), the two cotangent bundle varieties differ by a standard Mukai flop and this recovers an already known result. For \(k>1\) the varieties are related by a stratified Mukai flop and the \(k=2, N=4\) case was proved by Y. Kawamata [AMS/IP Stud. Adv. Math. 38, 285–294 (2006; Zbl 1137.14305)]. Reviewer: Dragos Deliu (Wien) Cited in 21 Documents MSC: 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 18E30 Derived categories, triangulated categories (MSC2010) 14M15 Grassmannians, Schubert varieties, flag manifolds 17B37 Quantum groups (quantized enveloping algebras) and related deformations Keywords:derived equivalences; cotangent bundle; Grassmannian; categorical actions Citations:Zbl 1137.14305; Zbl 1144.20001 PDFBibTeX XMLCite \textit{S. Cautis} et al., J. Reine Angew. Math. 675, 53--99 (2013; Zbl 1282.14034) Full Text: DOI arXiv