×

Derived equivalences for cotangent bundles of Grassmannians via categorical \(\mathfrak{sl}_2\) actions. (English) Zbl 1282.14034

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)].

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
PDFBibTeX XMLCite
Full Text: DOI arXiv