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Mirror Symmetry for Perverse Schobers from Birational Geometry

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Abstract

Perverse schobers are categorical analogs of perverse sheaves. Examples arise from varieties admitting flops, determined by diagrams of derived categories of coherent sheaves associated to the flop: in this paper we construct mirror partners to such schobers, determined by diagrams of Fukaya categories with stops, for examples in dimensions 2 and 3. Interpreting these schobers as supported on loci in mirror moduli spaces, we prove homological mirror symmetry equivalences between them. Our construction uses the coherent–constructible correspondence and a recent result of Ganatra et al. (Microlocal morse theory of wrapped fukaya categories. arXiv:1809.08807) to relate the schobers to certain categories of constructible sheaves. As an application, we obtain new mirror symmetry proofs for singular varieties associated to our examples, by evaluating the categorified cohomology operators of Bondal et al. (Selecta Math 24(1):85–143, 2018) on our mirror schobers.

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References

  1. Aspinwall, P.: A point’s point of view of stringy geometry. J. High Energy Phys. 1, 002 (2003). arXiv:hep-th/0203111

    Article  ADS  MathSciNet  Google Scholar 

  2. Ballard, M., Favero, D., Katzarkov, L.: Variation of geometric invariant theory quotients and derived categories. J. Reine Angew. Math. 2(746), 235 (2019). arXiv:1203.6643

    Article  MathSciNet  Google Scholar 

  3. Bergh, D., Lunts, V.A., Schnürer, O.M.: Geometricity for derived categories of algebraic stacks. Selecta Math. 22(4), 2535–2568 (2016). arXiv:1601.04465

    Article  MathSciNet  Google Scholar 

  4. Bondal, A.: Derived categories of toric varieties, convex and algebraic geometry, Oberwolfach conference reports, vol. 3. EMS Publishing House, pp. 284–286 (2006)

  5. Bodzenta, A., Bondal, A.: Flops and Spherical Functors. arXiv:1511.00665

  6. Bondal, A., Orlov, D.: Semiorthogonal Decomposition for Algebraic Varieties. arXiv:alg-geom/9506012

  7. Bondal, A., Kapranov, M., Schechtman, V.: Perverse schobers and birational geometry. Selecta Math. 24(1), 85–143 (2018). arXiv:1801.08286

    Article  MathSciNet  Google Scholar 

  8. Borisov, L.A., Chen, L., Smith, G.G.: The orbifold Chow ring of toric Deligne–Mumford stacks. J. Am. Math. Soc. 18, 193–215 (2005). arXiv:math/0309229

    Article  MathSciNet  Google Scholar 

  9. Bridgeland, T.: Flops and derived categories. Invent. Math. 147(3), 613–632 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  10. Bridgeland, T., King, A., Reid, M.: Mukai implies McKay: the McKay correspondence as an equivalence of derived categories. J. Am. Math. Soc. 14, 535–554 (2001). arXiv:math.AG/9908027

    Article  Google Scholar 

  11. Coates, T., Iritani, H., Jiang, Y., Segal, E.: \(K\)-theoretic and categorical properties of toric Deligne–Mumford stacks. Pure Appl. Math. Q. 11(2), 239–266 (2015). arXiv:1410.0027

    Article  MathSciNet  Google Scholar 

  12. Donovan, W.: Perverse schobers and wall crossing. Int. Math. Res. Not. 2019, rnx280 (2017). arXiv:1703.00592

    Google Scholar 

  13. Donovan, W.: Perverse schobers on Riemann surfaces: constructions and examples. Eur. J. Math. 5(3), 771–797 (2019). arXiv:1801.05319

    Article  MathSciNet  Google Scholar 

  14. Donovan, W., Wemyss, M.: Stringy Kähler moduli, mutation and monodromy. arXiv:1907.10891

  15. Fan, Y.-W., Hong, H., Lau, S.-C., Yau, S.-T.: Mirror of Atiyah flop in symplectic geometry and stability conditions. Adv. Theor. Math. Phys. 22(5) (2018). arXiv:1706.02942

  16. Fang, B.: Homological mirror symmetry is \(T\)-duality for \({\mathbb{P}}^n\). Commun. Math. Phys. 2(4), 719–742 (2008)

    MATH  Google Scholar 

  17. Fang, B., Liu, C.C., Treumann, D., Zaslow, E.: A categorification of Morelli’s theorem. Invent. Math. 186(1), 79–114 (2011). arXiv:1007.0053

    Article  ADS  MathSciNet  Google Scholar 

  18. Fantechi, B., Mann, E., Nironi, F.: Smooth toric Deligne–Mumford stacks. J. Reine Angew. Math. (Crelle’s J.) 648, 201 (2010). arXiv:0708.1254

    MathSciNet  MATH  Google Scholar 

  19. Gammage, B., Shende, V.: Mirror Symmetry for Very Affine Hypersurfaces. arXiv:1707.02959

  20. Ganatra, S., Pardon, J., Shende, V.: Sectorial Descent for Wrapped Fukaya Categories. arXiv: 1809.03427

  21. Ganatra, S., Pardon, J., Shende, V.: Microlocal Morse Theory of Wrapped Fukaya Categories. arXiv:1809.08807

  22. Geraschenko, A., Satriano, M.: Toric stacks I: the theory of stacky fans. Trans. Am. Math. Soc. 367(2), 1033 (2011). arXiv:1107.1906

    Article  MathSciNet  Google Scholar 

  23. Halpern-Leistner, D.: The derived category of a GIT quotient. J. Am. Math. Soc. 28(3), 871–912 (2015). arXiv:1203.0276

    Article  MathSciNet  Google Scholar 

  24. Halpern-Leistner, D., Shipman, I.: Autoequivalences of derived categories via geometric invariant theory. Adv. Math. 303, 1264–1299 (2016). arXiv:1303.5531

    Article  MathSciNet  Google Scholar 

  25. Harder, A., Katzarkov, L.: Perverse Sheaves of Categories and Some Applications. arXiv:1708.01181

  26. Huybrechts, D.: Fourier–Mukai Transforms in Algebraic Geometry, Oxford Mathematical Monographs. Oxford University Press, Oxford (2006)

    MATH  Google Scholar 

  27. Ishii, A., Ueda, K.: The special McKay correspondence and exceptional collection. Tohoku Math. J. 67(4), 585 (2011). arXiv:1104.2381

    Article  MathSciNet  Google Scholar 

  28. Kapranov, M., Schechtman, V.: Perverse sheaves over real hyperplane arrangements. Ann. Math. 183(2), 619–679 (2016). arXiv:1403.5800

    Article  MathSciNet  Google Scholar 

  29. Kapranov, M., Schechtman, V.: Perverse Schobers. arXiv:1411.2772

  30. Kashiwara, M., Schapira, P.: Sheaves on manifolds, with a chapter in French by Christian Houzel, Grundlehren der Mathematischen Wissenschaften, vol. 292. Springer, Berlin, 1990, x+512 pp

  31. Kawamata, Y.: Derived categories and birational geometry. In: Algebraic Geometry, Seattle 2005, Proceedings of 2005 Summer Research Institute. arXiv:0804.3150

  32. Kawamata, Y.: Log crepant birational maps and derived categories. J. Math. Sci. Univ. Tokyo 12, 211–231 (2005). arXiv:math/0311139

    MathSciNet  MATH  Google Scholar 

  33. Kawamata, Y.: Derived categories of toric varieties. Mich. Math. J. 54(3), 517–536 (2006). arXiv:math/0503102

    Article  MathSciNet  Google Scholar 

  34. Kawamata, Y.: Derived categories of toric varieties II. Mich. Math. J. 62(2), 353–363 (2013). arXiv:1201.3460

    Article  MathSciNet  Google Scholar 

  35. Kuwagaki, T.: The nonequivariant coherent-constructible correspondence for toric surfaces. J. Differ. Geom. 107(2), 373–393 (2017). arXiv:1507.05393

    Article  MathSciNet  Google Scholar 

  36. Kuwagaki, T.: The nonequivariant coherent-constructible correspondence for toric stacks. Duke Math. J. 169(11), 2125–2197 (2020). arXiv:1610.03214

    Article  MathSciNet  Google Scholar 

  37. Kuznetsov, A.: Semiorthogonal decompositions in algebraic geometry. Proc. ICM (Seoul, 2014) 2, 635–660 (2014). arXiv:1404.3143

    MathSciNet  MATH  Google Scholar 

  38. Kontsevich, M.: Symplectic Geometry of Homological Algebra. http://www.ihes.fr/~maxim/TEXTS/Symplectic_AT2009.pdf

  39. Nadler, D.: Arboreal singularities. Geom. Topol. 21(2), 1231–1274 (2017)

    Article  MathSciNet  Google Scholar 

  40. Nadler, D.: Wrapped Microlocal Sheaves on Pairs of Pants. arXiv:1604.00114

  41. Nadler, D.: Mirror symmetry for the Landau–Ginzburg A-model \(M={\mathbb{C}}^n\), \(W=z_1 \dots z_n\). Duke Math. J. 168(1), 1–84 (2019). arXiv:1601.02977

    Article  MathSciNet  Google Scholar 

  42. Nadler, D., Zaslow, E.: Constructible sheaves and the Fukaya category. J. Am. Math. Soc. 22(1), 233–286 (2009). arXiv:math/0604379

    Article  MathSciNet  Google Scholar 

  43. Neeman, A.: Triangulated Categories, Annals of Mathematics Studies, vol. 148. Princeton University Press, Princeton (2001)

    Book  Google Scholar 

  44. Orlov, D.: Projective bundles, monoidal transformations, and derived categories of coherent sheaves. Izv. Akad. Nauk SSSR Ser. Mat. 56(1992), 852–862; English transl., Russian Acad. Sci. Izv. Math. 41(1993), 133–141

  45. Scherotzke, S., Sibilla, N., Talpo, M.: Gluing Semi-orthogonal Decompositions. arXiv:1901.01257

  46. Sylvan, Z.: On partially wrapped Fukaya categories. J. Topol. 12(2), 372–441 (2019)

    Article  MathSciNet  Google Scholar 

  47. Zhou, P.: Lagrangian Skeleta of Hypersurfaces in \(({\mathbb{C}}^*)^n\). arXiv:1803.00320

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Acknowledgements

Both authors are grateful for conversations with A. Bondal and M. Kapranov at Kavli IPMU. The first author is grateful for discussions with T. Logvinenko, and thanks D. Auroux, T. Coates, and D. Nadler for helpful conversations at an early stage of the Project. The authors thank M. Romo for comments on the manuscript, and the organizers of the conference ‘Categorical and Analytic Invariants in Algebraic Geometry V’ at Osaka in 2018, where this collaboration started. Finally, the authors are grateful to an anonymous referee for many helpful comments. The authors acknowledge the support of WPI Initiative, MEXT, Japan, and of JSPS KAKENHI Grants JP16K17561 and JP18K13405 respectively which were held at Kavli IPMU, University of Tokyo. The first author is supported by the Yau MSC, Tsinghua University, and the Thousand Talents Plan.

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Communicated by H. T. Yau

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Donovan, W., Kuwagaki, T. Mirror Symmetry for Perverse Schobers from Birational Geometry. Commun. Math. Phys. 381, 453–490 (2021). https://doi.org/10.1007/s00220-020-03916-9

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