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Floer cohomology of the Chiang Lagrangian

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Abstract

We study holomorphic discs with boundary on a Lagrangian submanifold \(L\) in a symplectic manifold admitting a Hamiltonian action of a group \(K\) which has \(L\) as an orbit. We prove various transversality and classification results for such discs which we then apply to the case of a particular Lagrangian in \(\mathbf {C}\mathbf {P}^3\) first noticed by Chiang (Int Math Res Not 45:2437–2441, 2004). We prove that this Lagrangian has non-vanishing Floer cohomology if and only if the coefficient ring has characteristic 5, in which case, it generates the split-closed derived Fukaya category as a triangulated category.

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Notes

  1. If \(\bar{v}\) is in \(C\), then the sphere intersects \(Y_C\) at the smooth point comprising \(\bar{v}\) and the \((n-1)\)-fold point at \(v\); otherwise it intersects at the \(n\)-fold point at \(v\).

  2. Dangerous bend: \(D(F)\) no longer embeds in \(\widetilde{\mathbf {C}\mathbf {P}^{3}}\).

References

  1. 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 

  2. Abouzaid, M., Fukaya, K., Oh, Y.-G., Ohta, H., Ono, K.: In preparation

  3. Aluffi, P., Faber, C.: Linear orbits of \(d\)-tuples of points in \({ P}^1\). J. Reine Angew. Math. 445, 205–220 (1993)

    MathSciNet  MATH  Google Scholar 

  4. Atiyah, M.F., Bott, R., Shapiro, A.: Clifford modules. Topology 3(suppl. 1), 3–38 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  5. Auroux, D.: Mirror symmetry and \(T\)-duality in the complement of an anticanonical divisor. J. Gökova Geom. Topol. GGT 1, 51–91 (2007)

    MathSciNet  MATH  Google Scholar 

  6. Balmer, P., Schlichting, M.: Idempotent completion of triangulated categories. J. Algebra 236(2), 819–834 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bayer, A., Manin, Y.I.: (Semi)simple exercises in quantum cohomology. In: The Fano Conference, pp. 143–173. Univ. Torino, Turin (2004)

  8. Beauzamy, B., Bombieri, E., Enflo, P., Hugh L, M.: Products of polynomials in many variables. J. Number Theory 2, 219–245 (1990)

    Article  Google Scholar 

  9. Bedulli, L., Gori, A.: Homogeneous Lagrangian submanifolds. Comm. Anal. Geom. 16(3), 591–615 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Biran, P., Cornea, O.: A Lagrangian quantum homology. In: New Perspectives and Challenges in Symplectic Field Theory, Volume 49 of CRM Proceedings, Lecture Notes, pp. 1–44. American Mathematical Society, Providence, RI (2009)

  11. Biran, P., Cornea, O.: Lagrangian topology and enumerative geometry. Geom. Topol. 16(2), 963–1052 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chevalley, C.C.: The Algebraic Theory of Spinors. Columbia University Press, New York (1954)

    MATH  Google Scholar 

  13. Chiang, R.: New Lagrangian submanifolds of \({\mathbb{CP}}^n\). Int. Math. Res. Not. 45, 2437–2441 (2004)

    Article  Google Scholar 

  14. Cho, C.-H.: Holomorphic discs, spin structures, and Floer cohomology of the Clifford torus. Int. Math. Res. Not. 35, 1803–1843 (2004)

    Article  Google Scholar 

  15. Cho, C.-H.: Products of Floer cohomology of torus fibers in toric Fano manifolds. Comm. Math. Phys. 260(3), 613–640 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  16. Fukaya, K., Yong-Geun, O., Ohta, H., Ono, K.: Lagrangian Floer theory on compact toric manifolds. I. Duke Math. J. 151(1), 23–174 (2010)

    Article  MATH  Google Scholar 

  17. Fukaya, K., Yong-Geun, O.: Lagrangian Floer theory on compact toric manifolds II: bulk deformations. Selecta Math. (N.S.) 17(3), 609–711 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Globevnik, J.: Perturbation by analytic discs along maximal real submanifolds of \( {C}^N\). Math. Z. 217(2), 287–316 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hitchin, NJ.: Poncelet polygons and the Painlevé equations. In: Geometry and Analysis (Bombay, 1992), pp. 151–185. Tata Inst. Fund. Res., Bombay (1995)

  20. Hitchin, N.: A lecture on the octahedron. Bull. Lond. Math. Soc. 35(5), 577–600 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hitchin, N.: Vector bundles and the icosahedron. In: Vector bundles and Complex Geometry, Volume 522 of Contemporary Mathematics, pp. 71–87. American Mathematical Society, Providence, RI (2010)

  22. Kadeishvili, T.V.: The category of differential coalgebras and the category of \(A(\infty )\)-algebras. Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR 77, 50–70 (1985)

    MathSciNet  MATH  Google Scholar 

  23. Kassel, C.: A Künneth formula for the cyclic cohomology of \({ Z}/2\)-graded algebras. Math. Ann. 275(4), 683–699 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  24. Katz, S., Liu, C.-C.M.: Enumerative geometry of stable maps with Lagrangian boundary conditions and multiple covers of the disc. In: The interaction of finite-type and Gromov-Witten invariants (BIRS 2003), Volume 8 of Geometry and Topology Monographs, pp. 1–47. Geom. Topol. Publ., Coventry (2006)

  25. Kneebone, G.T., Semple, J.G.: Algebraic Projective Geometry. Clarendon Press, Oxford (1952)

  26. Lazzarini, L.: Existence of a somewhere injective pseudoholomorphic disc. Geom. Funct. Anal. 10(4), 829–862 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  27. Mori, S., Mukai, S.: Classification of Fano 3-folds with \(b_2\ge 2\). Manuscripta Mathematica 36, 147–162 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  28. Mukai, S., Umemura, H.: Minimal rational threefolds. In: Algebraic Geometry (Tokyo/Kyoto, 1982), Volume 1016 of Lecture Notes in Mathematics, pp. 490–518. Springer, Berlin (1983)

  29. Nakano, T.: On equivariant completions of \(3\)-dimensional homogeneous spaces of \({\rm SL} (2,{ C})\). Jpn. J. Math. (N.S.) 15(2), 221–273 (1989)

    MATH  Google Scholar 

  30. Yong-Geun, O.: Riemann–Hilbert problem and application to the perturbation theory of analytic discs. Kyungpook Math. J. 35(1), 39–75 (1995)

    MathSciNet  MATH  Google Scholar 

  31. Ritter, A., Smith, I.: The Open–Closed String Map Revisited. arXiv:1201.5880 (2012)

  32. Schwarzenberger, R.L.E.: Vector bundles on the projective plane. Proc. Lond. Math. Soc. 3(11), 623–640 (1961)

    Article  MathSciNet  Google Scholar 

  33. Seidel, P.: Homological Mirror Symmetry for the Quartic Surface. Mem. Am. Math. Soc. (to appear) arXiv:math/0310414 (2003)

  34. Seidel, P.: Fukaya Categories and Picard–Lefschetz Theory. Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich (2008)

  35. Seidel, P.: Abstract analogues of flux as symplectic invariants. Mémoires de la Soc. Math. Fr. 137 (2014)

  36. Sheridan, N.: On the Fukaya Category of a Fano Hypersurface in Projective Space. arXiv:1306.4143 (2013)

  37. Smith, I.: Floer cohomology and pencils of quadrics. Invent. Math. 189(1), 149–250 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  38. Taylor, J.L.: Several Complex Variables with Connections to Algebraic Geometry and Lie Groups, Volume 46 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (2002)

  39. Vekua, N.P.: Systems of Singular Integral Equations. P. Noordhoff Ltd., Groningen (1967). Translated from the Russian by A. G. Gibbs and G. M. Simmons. Edited by J. H. Ferziger

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Acknowledgments

J. E. would like to thank Jason Lotay for pointing out to him Hitchin’s papers on Platonic solids. Both authors would like to thank Ed Segal for helpful discussions on Clifford modules. Y. L. is supported by a Royal Society Fellowship. Figure 3 was produced using Fritz Obermeyer’s software Jenn3d.

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  1. University College London, London, UK

    Jonathan David Evans

  2. King’s College London, London, UK

    Yankı Lekili

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  1. Jonathan David Evans
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Evans, J.D., Lekili, Y. Floer cohomology of the Chiang Lagrangian. Sel. Math. New Ser. 21, 1361–1404 (2015). https://doi.org/10.1007/s00029-014-0171-9

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