Kyoto Journal of Mathematics

Seidel’s long exact sequence on Calabi-Yau manifolds

Yong-Geun Oh

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Abstract

In this paper, we generalize construction of Seidel’s long exact sequence of Lagrangian Floer cohomology to that of compact Lagrangian submanifolds with vanishing Malsov class on general Calabi-Yau manifolds. We use the framework of anchored Lagrangian submanifolds and some compactness theorem of smooth J-holomorphic sections of Lefschetz Hamiltonian fibration for a generic choice of J. The proof of the latter compactness theorem involves a study of proper pseudoholomorphic curves in the setting of noncompact symplectic manifolds with cylindrical ends.

Article information

Source
Kyoto J. Math., Volume 51, Number 3 (2011), 687-765.

Dates
First available in Project Euclid: 1 August 2011

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1312205244

Digital Object Identifier
doi:10.1215/21562261-1299936

Mathematical Reviews number (MathSciNet)
MR2824005

Zentralblatt MATH identifier
1230.53079

Subjects
Primary: 53D37: Mirror symmetry, symplectic aspects; homological mirror symmetry; Fukaya category [See also 14J33] 53D40: Floer homology and cohomology, symplectic aspects

Citation

Oh, Yong-Geun. Seidel’s long exact sequence on Calabi-Yau manifolds. Kyoto J. Math. 51 (2011), no. 3, 687--765. doi:10.1215/21562261-1299936. https://projecteuclid.org/euclid.kjm/1312205244


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References

  • [Bo] G. Bourgeois, A Morse-Bott approach to contact homology, Ph.D. dissertation, Stanford University, ProQuest, Ann Arbor, Mich., 2002.
  • [EGH] Y. Eliashberg, A. Givental, and H. Hofer, “Introduction to symplectic field theory” in GAFA 2000 (Tel Aviv, 1999), Geom. Funct. Anal. 2000, special vol., part II, 560–673.
  • [En] M. Entov, K-area, Hofer metric and geometry of conjugacy classes in Lie groups, Invent. Math. 146 (2000), 93–141.
  • [Fl] A. Floer, Morse theory for Lagrangian intersections, J. Differential Geom. 28 (1988), 513–547.
  • [Fu] K. Fukaya, “Floer homology and mirror symmetry, II” in Minimal Surfaces, Geometric Analysis and Symplectic Geometry (Baltimore, 1999), Adv. Stud. Pure Math. 34, Math. Soc. Japan, Tokyo, 2002, 31–127.
  • [FO+1] K. Fukaya, Y.-G. Oh, H. Ohta, and K. Ono, Lagrangian Intersection Floer Theory: Anomaly and Obstruction, I, AMS/IP Stud. Adv. Math. 46.1, AMS/Int. Press, Amer. Math. Soc., Providence, 2009; II, AMS/IP Stud. Adv. Math. 46.2, AMS/Int. Press, Amer. Math. Soc., Providence, 2009.
  • [FO+2] K. Fukaya, Y.-G. OH, H. Ohta, and K. Ono, “Anchored Lagrangian submanifolds and their Floer theory” in Mirror Symmetry and Tropical Geometry 527, Amer. Math. Soc., Providence, 2010, 15–54.
  • [FO+3] K. Fukaya and Y.-G. Oh, Lagrangian intersection Floer theory: Anomaly and obstruction, preprint, 2000.
  • [FO+4] K. Fukaya, Y.-G. OH, H. Ohta, and K. Ono, Lagrangian surgery and metamorphosis of pseudoholomorphic polygons, preprint, 2007.
  • [FOn] K. Fukaya and K. Ono, Arnold conjecture and Gromov-Witten invariant, Topology 38 (1999), 933–1048.
  • [Gr] M. Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), 307–347.
  • [GLS] V. Guillemin, E. Lerman, and S. Sternberg, Symplectic Fibrations and Multiplicity Diagrams, Cambridge Univ. Press, Cambridge, 1996.
  • [H] H. Hofer, Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three, Invent. Math. 114 (1993), 515–563.
  • [HL] S. Hu and F. Lalonde, Homological Lagrangian monodromy, preprint, arXiv:0912.1325v3 [math. SG]
  • [KL] S. Katz and M. Liu, Enumerative geometry of stable maps with Lagrangian boundary conditions and multiple covers of the disc, Adv. Theor. Math. Phys. 5 (2001), 1–49.
  • [K1] M. Kontsevich, “Homological algebra of mirror symmetry” in Proceedings of the International Congress of Mathematicians, Vols. 1, 2(Zürich, 1994), Birkhäuser, Basel, 1995, 120–139.
  • [K2] M. Kontsevich, Lecture at Rutgers University, November 11, 1996.
  • [MS] D. McDuff and D. Salamon, J-Holomorphic Curves and Symplectic Topology, Colloq. Publ. 52, Amer. Math. Soc., Providence, 2004.
  • [Mo] K. Mohnke, How to (symplectically) thread the eye of a (Lagrangian) needle, preprint, arXiv:math/0106139v4 [math.SG]
  • [Oh1] Y.-G. Oh, Floer cohomology of Lagrangian intersections and pseudoholomorphic discs, I, Comm. Pure Appl. Math. 46 (1993), 949–994; II, 995–1012; Addendum, Comm. Pure Appl. Math. 48 (1995), 1299–1302.
  • [Oh2] Y.-G. Oh, Symplectic topology as the geometry of action functional, I, J. Differential Geom. 46 (1997), 499–577.
  • [Oh3] Y.-G. Oh, Spectral invariants, analysis of the Floer moduli space, and geometry of Hamiltonian diffeomorphism group, Duke Math. J. 130 (2005), 199–295.
  • [OF] Y.-G. Oh and K. Fukaya, “Floer homology in symplectic geometry and in mirror symmetry” in International Congress of Mathematicians, Vol. II, Eur. Math. Soc., Zürich, 2006, 879–905.
  • [OZ] Y.-G. Oh and K. Zhu, Floer trajectories with immersed nodes and scale-dependent gluing, preprint, arXiv:0711.4187v3 [math. SG]
  • [RS] J. Robbin and D. Salamon, The Maslov index for paths, Topology 32 (1993), 827–844.
  • [RT] Y. Ruan and G. Tian, A mathematical theory of quantum cohomology, J. Differential Geom. 42 (1995), 259–367.
  • [Se1] P. Seidel, Lagrangian two-spheres can be symplectically knotted, J. Differential Geom. 52 (1999), 145–171.
  • [Se2] P. Seidel, Graded Lagrangian submanifolds, Bull. Soc. Math. France 128 (2000), 103–149.
  • [Se3] P. Seidel, A long exact sequence for symplectic Floer cohomology, Topology 42 (2003), 1003–1063.
  • [Se4] P. Seidel, Fukaya Categories and Picard-Sefschetz Theory, Zur. Lect. Adv. Math., Eur. Math. Soc., Zürich, 2008.