Skip to main content
SpringerLink
Log in

Arithmetic mirror symmetry for genus 1 curves with n marked points

  • Published:
Selecta Mathematica Aims and scope Submit manuscript

Abstract

We establish a \({\mathbb {Z}}[[t_1,\ldots , t_n]]\)-linear derived equivalence between the relative Fukaya category of the 2-torus with n distinct marked points and the derived category of perfect complexes on the n-Tate curve. Specialising to \(t_1= \cdots =t_n=0\) gives a \({\mathbb {Z}}\)-linear derived equivalence between the Fukaya category of the n-punctured torus and the derived category of perfect complexes on the standard (Néron) n-gon. We prove that this equivalence extends to a \({\mathbb {Z}}\)-linear derived equivalence between the wrapped Fukaya category of the n-punctured torus and the derived category of coherent sheaves on the standard n-gon. The corresponding results for \(n=1\) were established in Lekili and Perutz (Arithmetic mirror symmetry for the 2-torus (preprint) arXiv:1211.4632, 2012).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abouzaid, M., Auroux, D., Efimov, A., Katzarkov, L., Orlov, D.: Homological mirror symmetry for punctured spheres. J. Am. Math. Soc. 26(4), 1051–1083 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  2. Abouzaid, M.: A geometric criterion for generating the Fukaya category. Publ. Math. Inst. Hautes Études Sci. 112, 191–240 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Abouzaid, M., Seidel, P.: An open string analogue of Viterbo functoriality. Geom. Topol. 14(2), 627–718 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Abouzaid, M., Smith, I.: Homological mirror symmetry for the 4-torus. Duke Math. J. 152(3), 373–440 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Barvinok, A.: Integer points in polyhedra. In: Zürich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich (2008)

  6. Bass, H.: On the ubiquity of Gorenstein rings. Math. Z. 82, 8–28 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bocklandt, R.: Noncommutative mirror symmetry for punctured surfaces, with an appendix by Mohammed Abouzaid. Trans. Am. Math. Soc. 368(1), 429–469 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bondal, A., Van den Bergh, M.: Generators and representability of functors in commutative and noncommutative geometry. Mosc. Math. J. 3(1), 1–36 (2003)

    MathSciNet  MATH  Google Scholar 

  9. Burban, I., Drozd, Yu.: Coherent sheaves on rational curves with simple double points and transversal intersections. Duke Math. J. 121(2), 189–229 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. Brion, M.: Points entiers dans les polyèdres convexes. Ann. Sci. École Norm. Sup. (4) 21(4), 653–663 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  11. Deligne, P., Mumford, D.: The irreducibility of the space of curves of given genus. Inst. Hautes Études Sci. Publ. Math. 36, 75–109 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  12. Deligne, P., Rapoport, M.: Les schémas de modules de courbes elliptiques. In: Modular Functions of One Variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Lecture Notes in Math., vol. 349, pp. 143–316. Springer, Berlin (1973)

  13. Dieudonné, J., Grothendieck, A.: EGA III, Étude Cohomologique des Faisceaux Cohérents (Première Partie). Publ. Math. Inst. Hautes Études Sci. 11, 167 (1961)

  14. Dwyer, W.G., Greenless, J.P.C.: Complete modules and torsion modules. Am. J. Math. 124(1), 199–220 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  15. Efimov, A.: Homological mirror symmetry for curves of higher genus. Adv. Math. 230(2), 493–530 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Efimov, A.: Formal completion of a category along a subcategory. preprint (2010), arXiv:1006.4721

  17. Etgü, T., Lekili, Y.: Koszul duality patterns in Floer theory. preprint (2015), To appear in Geom. Topol. arXiv:1502.07922

  18. Farb, B., Margalit, D.: A Primer on Mapping Class Groups. Princeton Mathematical Series 49. Princeton University Press, Princeton, NJ (2012)

    MATH  Google Scholar 

  19. Fukaya, K., Oh, Y.-G., Ohta, H., Ono, K.: Lagrangian surgery and holomorphic discs. https://www.math.kyoto-u.ac.jp/~fukaya/Chapter10071117

  20. Ganatra, S.: Symplectic Cohomology and Duality for the Wrapped Fukaya Category. MIT PhD thesis (2012)

  21. Gross, M.: Chapter 8 of dirichlet branes and mirror symmetry. In: Clay Mathematics Monographs, 4. Amer. Math. Soc., Providence, RI; Clay Mathematics Institute, Cambridge, MA (2009)

  22. Haiden, F., Katzarkov, L., Kontsevich, M.: Flat surfaces and stability structures. Preprint (2014), arXiv:1409.8611

  23. Jarvis, T.J.: Torsion free sheaves and moduli of generalized spin-curves. Compos. Math. 110(3), 291–333 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  24. Keating, A.: Homological mirror symmetry for hypersurface cusp singularities. Preprint (2015), arXiv:1510.08911

  25. Keller, B.: Deriving DG categories. Ann. Sci. École Norm. Sup. (4) 27(1), 63–102 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  26. Keller, B.: Introduction to \(A_\infty \)-algebras and modules. Homology Homotopy Appl. 3(1), 1–35 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  27. Keller, B.: Derived invariance of higher structures on Hochschild complex. Preprint (2003), http://webusers.imj-prg.fr/~bernhard.keller/publ/dih

  28. Kontsevich, M.: Homological algebra of mirror symmetry. In: Proceedings of the International Congress of Mathematicians, vol. 1. 2 (Zürich, 1994). Birkhäuser, Basel (1995)

  29. Kontsevich, M.: Symplectic geometry of homological algebra. Preprint (2009), http://www.ihes.fr/~maxim/TEXTS/Symplectic_AT2009

  30. Lee, H.-M.: Homological mirror symmetry for open Riemann surfaces from pair-of-pants decompositions. Preprint (2016), arXiv:1608.04473

  31. Lekili, Y., Perutz, T.: Arithmetic mirror symmetry for the 2-torus. Preprint (2012), arXiv:1211.4632

  32. Lekili, Y., Perutz, T.: Fukaya categories of the torus and Dehn surgery. Proc. Natl. Acad. Sci. USA 108(20), 8106–8113 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  33. Lekili, Y., Polishchuk, A.: A modular compactification of \({\cal M}_{1,n}\) from \(A_\infty \) structures. Preprint (2014), arXiv:1408.0611

  34. Lu, D.M., Palmieri, J.H., Wu, Q.S., Zhang, J.J.: Koszul equivalences in \(A_\infty \)-algebras. N. Y. J. Math. 14, 325–378 (2008)

    MathSciNet  MATH  Google Scholar 

  35. Nadler, D., Zaslow, E.: Constructible sheaves and the Fukaya category. J. Am. Math. Soc. 22, 233–286 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  36. Orlov, D.: Formal completions and idempotent completions of triangulated categories of singularities. Adv. Math. 226(1), 206–217 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  37. Orlov, D.: Remarks on generators and dimensions of triangulated categories. Mosc. Math. J. 9(1), 153–159 (2009)

    MathSciNet  MATH  Google Scholar 

  38. Polishchuk, A., Zaslow, E.: Categorical mirror symmetry: the elliptic curve. Adv. Theor. Math. Phys. 2, 443–470 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  39. Polishchuk, A.: Massey and Fukaya products on elliptic curves. Adv. Theor. Math. Phys. 4, 1187–1207 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  40. Polishchuk, A.: Extensions of homogeneous co-ordinate rings to \(A_\infty \)-algebras. Homol. Homotopy Appl. 5, 407–421 (2003)

    Article  MATH  Google Scholar 

  41. Polishchuk, A.: \(A_\infty \)-structures on an elliptic curve. Commun. Math. Phys. 247, 527–551 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  42. Polishchuk, A.: Moduli of curves as moduli of \(A_\infty \)-structures. Preprint (2013), arXiv:1312.4636

  43. Polishchuk, A.: Moduli of curves with nonspecial divisors and relative moduli of \(A_\infty \)-structures. Preprint (2015), arXiv:1511.03797

  44. Seidel, P.: Fukaya categories and deformations. In: Proceedings of the International Congress of Mathematics, vol. II (Beijing, 2002), pp. 351–360. Higher Ed. Press, Beijing (2002)

  45. Seidel, P.: Homological mirror symmetry for the quartic surface. Mem. Am. Math. Soc. 236(1116), vi+129 pp. (2015)

  46. Seidel, P.: Homological mirror symmetry for the genus two curve. J. Algebraic Geom. 20(4), 727–769 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  47. Seidel, P.: A biased view of symplectic cohomology, in Current developments in mathematics. International Press, Somerville (2008)

    Google Scholar 

  48. Seidel, P.: Fukaya categories and Picard–Lefschetz theory. In: Zürich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich (2008)

  49. Seidel, P.: A long exact sequence for symplectic Floer cohomology. Topology 42(5), 1003–1063 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  50. Seidel, P.: Symplectic Floer homology and the mapping class group. Pac. J. Math. 206(1), 219–229 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  51. Seidel, P.: Graded lagrangian submanifolds. Bull. Soc. Math. Fr. 128, 103–149 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  52. Sibilla, N., Treumann, D., Zaslow, E.: Ribbon graphs and mirror symmetry. Sel. Math. (N.S.) 20(4), 979–1002 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  53. Smyth, D.I.: Modular compactifications of the space of pointed elliptic curves I. Compos. Math. 147(3), 877–913 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  54. Thomason, R.: The classification of triangulated subcategories. Compos. Math. 105(1), 1–27 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  55. Toën, B.: The homotopy theory of dg-categories and derived Morita theory. Invent. Math. 167(3), 615–667 (2007)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

Y.L. is supported in part by the Royal Society and the NSF Grant DMS-1509141. A.P. is supported in part by the NSF Grant DMS-1400390.

Author information

Authors and Affiliations

  1. King’s College London, London, United Kingdom

    Yankı Lekili

  2. University of Oregon, Eugene, OR, USA

    Alexander Polishchuk

Authors
  1. Yankı Lekili
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alexander Polishchuk
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexander Polishchuk.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lekili, Y., Polishchuk, A. Arithmetic mirror symmetry for genus 1 curves with n marked points. Sel. Math. New Ser. 23, 1851–1907 (2017). https://doi.org/10.1007/s00029-016-0286-2

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00029-016-0286-2

Mathematics Subject Classification

Search

Navigation