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European Journal of Mathematics

, Volume 5, Issue 3, pp 771–797 | Cite as

Perverse schobers on Riemann surfaces: constructions and examples

  • Will DonovanEmail author
Research Article
  • 35 Downloads

Abstract

This note studies perverse sheaves of categories, or schobers, on Riemann surfaces, following ideas of Kapranov and Schechtman (Perverse schobers, arXiv:1411.2772, 2014). For certain wall crossings in geometric invariant theory, we construct a schober on the complex plane, singular at each imaginary integer. We use this to obtain schobers for standard flops: in the threefold case, we relate these to a further schober on a partial compactification of a stringy Kähler moduli space, and suggest an application to mirror symmetry.

Keywords

Perverse sheaves Perverse schobers Derived categories Riemann surfaces Geometric invariant theory Flops Mirror symmetry 

Mathematics Subject Classification

14F05 14E05 14J33 14L24 18E30 

Notes

Acknowledgements

I am grateful to Mikhail Kapranov for inspiring conversations. I thank Alexey Bondal, Yukari Ito, Alastair King, Sven Meinhardt, Ed Segal, and Michael Wemyss for useful discussions, and Jacopo Stoppa and Barbara Fantechi for their hospitality and interest in my work at SISSA, Trieste. I am grateful to an anonymous referee, and to Pierre Schapira, for helpful comments. Finally, I thank the organizers of the 2016 Easter Island workshop on algebraic geometry for the opportunity to attend.

References

  1. 1.
    Addington, N.: New derived symmetries of some hyperkähler varieties. Algebraic Geom. 3(2), 223–260 (2016). arXiv:1112.0487 MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Anno, R., Logvinenko, T.: Spherical DG functors. J. Eur. Math. Soc. 19(9), 2577–2656 (2017). arXiv:1309.5035 MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Aspinwall, P.S.: A point’s point of view of stringy geometry (2002). arXiv:hep-th/0203111
  4. 4.
    Ballard, M., Favero, D., Katzarkov, L.: Variation of geometric invariant theory quotients and derived categories. J. Reine Angew. Math. https://doi.org/10.1515/crelle-2015-0096. arXiv:1203.6643
  5. 5.
    Beilinson, A.A.: How to glue perverse sheaves. In: Manin, Yu.I. (ed.) K-theory, Arithmetic and Geometry. Lecture Notes in Mathematics, vol. 1289, pp. 42–51. Springer, Berlin (1987)Google Scholar
  6. 6.
    Beilinson, A.A., Bernstein, J., Deligne, P.: Faisceaux pervers. Astérisque, vol. 100. Société Mathématique de France, Paris (1982). https://webusers.imj-prg.fr. Accessed 20 Dec 2017
  7. 7.
    Bodzenta, A., Bondal, A.: Flops and spherical functors (2015). arXiv:1511.00665
  8. 8.
    Bondal, A., Orlov, D.: Semiorthogonal decomposition for algebraic varieties (1995). arXiv:alg-geom/9506012
  9. 9.
    Bondal, A., Kapranov, M., Schechtman, V.: Perverse schobers and birational geometry. Selecta Math. (N.S.) 24(1), 85–143 (2018). arXiv:1801.08286 MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Chan, K., Pomerleano, D., Ueda, K.: Lagrangian torus fibrations and homological mirror symmetry for the conifold. Commun. Math. Phys. 341(1), 135–178 (2016). arXiv:1305.0968 MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Coates, T., Iritani, H., Jiang, Y.: The crepant transformation conjecture for toric complete intersections. Adv. Math. 329, 1002–1087 (2018). arXiv:1410.0024 MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Donovan, W.: Perverse schobers and wall crossing. Int. Math. Res. Not. IMRN. https://doi.org/10.1093/imrn/rnx280. arXiv:1703.00592
  13. 13.
    Donovan, W., Segal, E.: Mixed braid group actions from deformations of surface singularities. Comm. Math. Phys. 335(1), 497–543 (2014). arXiv:1310.7877 MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Donovan, W., Wemyss, M.: Noncommutative deformations and flops. Duke Math. J. 165(8), 1397–1474 (2016). arXiv:1309.0698 MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Donovan, W., Wemyss, M.: Twists and braids for general 3-fold flops. J. Eur. Math. Soc. (in press). arXiv:1504.05320
  16. 16.
    Dyckerhoff, T., Kapranov, M., Schechtman, V., Soibelman, Y.: Perverse schobers on surfaces and Fukaya categories with coefficients (in preparation)Google Scholar
  17. 17.
    Fabel, P.: The mapping class group of a disk with infinitely many holes. J. Knot Theory Ramifications 15(1), 21–29 (2006). arXiv:math/0303042 MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Fan, Y.-W., Hong, H., Lau, S.-C., Yau, S.-T.: Mirror of Atiyah flop in symplectic geometry and stability conditions (2017). arXiv:1706.02942
  19. 19.
    Gelfand, S., MacPherson, R., Vilonen, K.: Perverse sheaves and quivers. Duke Math. J. 83(3), 621–643 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Halpern-Leistner, D.: The derived category of a GIT quotient. J. Amer. Math. Soc. 28(3), 871–912 (2015). arXiv:1203.0276 MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Halpern-Leistner, D., Sam, S.V.: Combinatorial constructions of derived equivalences (2016). arXiv:1601.02030
  22. 22.
    Halpern-Leistner, D., Shipman, I.: Autoequivalences of derived categories via geometric invariant theory. Adv. Math. 303, 1264–1299 (2016). arXiv:1303.5531 MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Harder, A., Katzarkov, L.: Perverse sheaves of categories and some applications (2017). arXiv:1708.01181
  24. 24.
    Huybrechts, D.: Fourier–Mukai Transforms in Algebraic Geometry. Oxford Mathematical Monographs. Oxford University Press, Oxford (2006)zbMATHGoogle Scholar
  25. 25.
    Kapranov, M., Schechtman, V.: Perverse sheaves over real hyperplane arrangements. Ann. Math. 183(2), 619–679 (2016). arXiv:1403.5800 MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Kapranov, M., Schechtman, V.: Perverse schobers (2014). arXiv:1411.2772
  27. 27.
    Kapranov, M., Schechtman, V.: Perverse sheaves and graphs on surfaces (2016). arXiv:1601.01789
  28. 28.
    Kashiwara, M.: On the maximally overdetermined system of linear differential equations. I. Publ. Res. Inst. Math. Sci. IMRN 10, 563–579 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Kashiwara, M.: The Riemann–Hilbert problem for holonomic systems. Publ. Res. Inst. Math. Sci. IMRN 20(2), 319–365 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Katzarkov, L., Pandit, P., Spaide, T.: Calabi–Yau structures, spherical functors, and shifted symplectic structures (2017). arXiv:1701.07789
  31. 31.
    Mebkhout, Z.: Une autre équivalence de catégories. Compositio Math. 51(1), 63–88 (1984)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Nadler, D.: Mirror symmetry for the Landau–Ginzburg A-model \(M=\mathbb{C}^n\), \(W=z_1 \cdots z_n\) (2016). arXiv:1601.02977
  33. 33.
    Segal, E.: All autoequivalences are spherical twists. Int. Math. Res. Not. IMRN 2018(10), 3137–3154 (2018). arXiv:1603.06717 MathSciNetzbMATHGoogle Scholar
  34. 34.
    Toda, Y.: On a certain generalization of spherical twists. Bull. Soc. Math. France 135(1), 119–134 (2007). arXiv:math/0603050 MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Toda, Y.: Stability conditions and crepant small resolutions. Trans. Amer. Math. Soc. 360(11), 6149–6178 (2008). arXiv:math/0512648 MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Toda, Y.: Non-commutative width and Gopakumar–Vafa invariants. Manuscripta Math. 148(3–4), 521–533 (2015). arXiv:1411.1505 MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Yau Mathematical Sciences CenterTsinghua UniversityHaidian District, BeijingChina

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