Twisted polytope sheaves and coherent–constructible correspondence for toric varieties
- 9 Downloads
Given a smooth projective toric variety \(X_\Sigma \) of complex dimension n, Fang–Liu–Treumann–Zaslow (Invent Math 186(1):79–114, 2011) showed that there is a quasi-embedding of the differential graded (dg) derived category of coherent sheaves \(Coh(X_\Sigma )\) into the dg derived category of constructible sheaves on a torus \(Sh(T^n, \Lambda _\Sigma )\). Recently, Kuwagaki (The nonequivariant coherent-constructible correspondence for toric stacks, 2016. arXiv:1610.03214) proved that the quasi-embedding is a quasi-equivalence, and generalized the result to toric stacks. Here we give a different proof in the smooth projective case, using non-characteristic deformation of sheaves to find twisted polytope sheaves that co-represent the stalk functors.
Mathematics Subject Classification53D37 (Homological Mirror Symmetry)
It is a pleasure to thank Xin Jin, Linhui Shen, Lei Wu, Elden Elmanto, Dima Tamarkin for many helpful discussions, and David Treumann and David Nadler for their interests in this work. I am grateful for Elden for carefully reading the draft and giving many useful comments. I am greatly indebted to my advisor Eric Zaslow for inspirations and encouragements (and patience!). The discussion with David Treumann at IAS inspired the current approach, which uses the twisted polytope sheaves to corepresent the stalk functor. I am thankful for the referee for many useful suggestions.
- 1.Bondal, A.: Derived categories of toric varieties. In: Convex and Algebraic Geometry, Oberwolfach Conference Reports. EMS Publishing House, vol. 3, pp. 284–286 (2006)Google Scholar
- 13.Keller, B.: On differential graded categories. arXiv:math/0601185
- 14.Kashiwara, M., Schapira, P.: Sheaves on Manifolds. Springer, Berlin, Heidelberg (1990). https://doi.org/10.1007/978-3-662-02661-8
- 17.Kuwagaki, T.: The nonequivariant coherent-constructible correspondence for toric stacks (2016). arXiv:1610.03214
- 19.Nadler, D.: Wrapped microlocal sheaves on pairs of pants. arXiv:1604.00114 [math.SG]
- 22.Schapira, P.: A short review on microlocal sheaf theory. https://webusers.imj-prg.fr/~pierre.schapira/lectnotes/MuShv.pdf
- 23.Shende, V., Treumann, D., Williams, H.: On the combinatorics of exact Lagrangian surfaces. arXiv:1603.07449
- 24.Treumann, D.: Remarks on the nonequivariant coherent-constructible correspondence for toric varieties (2010). arXiv:1006.5756
- 25.Vaintrob, D.: Microlocal mirror symmetry on the torus, available at the authors homepageGoogle Scholar
- 26.Zhou, P.: Variation of GIT quotients and constructible sheaves. In preparationGoogle Scholar
- 27.Zhou, P.: Sheaf Quantization of Legendrian Isotopy. arxiv:1804.08928