Duke Mathematical Journal

Lagrangian Floer theory on compact toric manifolds, I

Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono

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Abstract

We introduced the notion of weakly unobstructed Lagrangian submanifolds and constructed their potential function ($\mathfrak{PO}$) purely in terms of $A$-model data in [FOOO3]. In this article, we carry out explicit calculations involving $\mathfrak{PO}$ on toric manifolds and study the relationship between this class of Lagrangian submanifolds with the earlier work of Givental [G1], which advocates that the quantum cohomology ring is isomorphic to the Jacobian ring of a certain function, called the Landau-Ginzburg superpotential. Combining this study with the results from [FOOO3], we also apply the study to various examples to illustrate its implications to symplectic topology of Lagrangian fibers of toric manifolds. In particular, we relate it to the Hamiltonian displacement property of Lagrangian fibers and to Entov-Polterovich's symplectic quasi-states

Article information

Source
Duke Math. J. Volume 151, Number 1 (2010), 23-175.

Dates
First available in Project Euclid: 31 December 2009

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1262271306

Digital Object Identifier
doi:10.1215/00127094-2009-062

Mathematical Reviews number (MathSciNet)
MR2573826

Zentralblatt MATH identifier
05663374

Subjects
Primary: 53D12: Lagrangian submanifolds; Maslov index 53D40: Floer homology and cohomology, symplectic aspects
Secondary: 14J45: Fano varieties 14J32: Calabi-Yau manifolds

Citation

Fukaya, Kenji; Oh, Yong-Geun; Ohta, Hiroshi; Ono, Kaoru. Lagrangian Floer theory on compact toric manifolds, I. Duke Math. J. 151 (2010), no. 1, 23--175. doi:10.1215/00127094-2009-062. http://projecteuclid.org/euclid.dmj/1262271306.


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References

  • M. Abouzaid, Homogeneous coordinate rings and mirror symmetry for toric varieties, Geom. Topol. 10 (2006), 1097--1157.
  • M. Audin, Torus Actions on Symplectic Manifolds, 2nd ed., Progr. Math. 93, Birkhäuser, Basel, 2004.
  • D. Auroux, Mirror symmetry and T-duality in the complement of an anticanonical divisor, J. Gökova Geom. Topol. GGT 1 (2007), 51--91.
  • D. Auroux, L. Katzarkov, and D. Orlov, Mirror symmetry for weighted projective planes and their noncommutative deformations, Ann. of Math. (2) 167 (2008), 867--943.
  • V. V. Batyrev, ``Quantum cohomology rings of toric manifolds'' in Journées de géométrie algébrique d'Orsay (Orsay, France, 1992), Astérisque 218, Soc. Math. France, Montrouge, 1993, 9--34.
  • —, Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. Algebraic Geom. 3 (1994), 493--535.
  • S. Bosch, U. GüNtzer, and R. Remmert, Non-Archimedean Analysis, Grundlehren Math. Wiss. 261, Springer, Berlin, 1984.
  • K. Chan and N. C. Leung, Mirror symmetry for toric Fano manifolds via SYZ transformations, preprint.
  • Yu. V. Chekanov, Lagrangian intersections, symplectic energy, and areas of holomorphic curves, Duke Math. J. 95 (1998), 213--226.
  • C.-H. Cho, Non-displaceable Lagrangian submanifolds and Floer cohomology with non-unitary line bundle, J. Geom. Phys. 58 (2008), 1465--1476.
  • C.-H. Cho and Y.-G. Oh, Floer cohomology and disc instantons of Lagrangian torus fibers in Fano toric manifolds, Asian J. Math. 10 (2006), 773--814.
  • K. Cieliebak and D. Salamon, Wall crossing for symplectic vortices and quantum cohomology, Math. Ann. 335 (2006), 133--192.
  • D. A. Cox and S. Katz, Mirror Symmetry and Algebraic Geometry, Math. Surveys Monogr. 68, Amer. Math. Soc., Providence, 1999.
  • T. Delzant, Hamiltoniens périodiques et image convexes de l'application moment, Bull. Soc. Math. France 116 (1988), 315--339.
  • D. Eisenbud, Commutative Algebra, Grad. Texts in Math. 150, Springer, Berlin, 1994.
  • M. Entov and L. Polterovich, Quasi-states and symplectic intersections, Comment. Math. Helv. 81 (2006), 75--99.
  • —, ``Symplectic quasi-states and semi-simplicity of quantum homology'' in Toric Topology (Osaka, Japan, 2006), Contemp. Math. 460, Amer. Math. Soc., Providence, 2008, 47--70.
  • —, Rigid subsets of symplectic manifolds, Compos. Math. 145 (2009), 773--826.
  • A. Floer, Morse theory for Lagrangian intersections, J. Differential Geom. 28 (1988), 513--547.
  • K. Fukaya, ``Morse homotopy, $A^\infty$-category, and Floer homologies'' in Proceedings of GARC Workshop on Geometry and Topology '93 (Seoul, 1993), Lecture Notes Ser. 18, Seoul National University, Seoul, 1993.
  • —, ``Floer homology for families --.-A progress report'' in Integrable Systems, Topology, and Physics (Tokyo, 2000), Contemp. Math. 309, Amer. Math. Soc., Providence, 2002, 33--68.
  • —, Mirror symmetry of abelian varieties and multi-theta functions, J. Algebraic Geom. 11 (2002), 393--512.
  • —, ``Application of Floer homology of Lagrangian submanifolds to symplectic topology'' in Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topology (Montreal, 2004), NATO Sci. Ser. II Math. Phys. Chem. 217, Springer, Dordrecht, 2006, 231--276.
  • —, ``Differentiable operand, Kuranishi correspondence, and foundation of topological field theories based on pseudo-holomorphic curve,'' to appear in Arithmetic and Geometry around Quantization, Prog. Math. 279, Birkhäuser, Boston.
  • K. Fukaya, Y.-G. Oh, H. Ohta, and K. Ono, Lagrangian intersection Floer theory-anomaly and obstruction, preprint, 2000.
  • —, Lagrangian intersection Floer theory-anomaly and obstruction, preprint, 2006, 2007.
  • —, Lagrangian Intersection Floer Theory-Anomaly and Obstruction, AMS/IP Stud. Adv. Math. 46, Amer. Math. Soc., Providence, 2009.
  • —, ``Canonical models of filtered $A_\infty$-algebras and Morse complexes'' in New Perspectives and Challenges in Symplectic Field Theory, CRM Proc. Lecture Notes 49, Amer. Math. Soc., Providence, 2009, 201--228.
  • —, Lagrangian Floer theory on compact toric manifolds, II: Bulk deformations, preprint.
  • K. Fukaya and K. Ono, Arnold conjecture and Gromov-Witten invariant, Topology 38 (1999), 933--1048.
  • W. Fulton, Introduction to Toric Varieties, Ann. of Math. Stud. 131, Princeton Univ. Press, Princeton, 1993.
  • A. B. Givental, ``Homological geometry and mirror symmetry'' in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), Birkhäuser, Basel, 1995, 472--480.
  • —, Equivariant Gromov-Witten invariants, Internat. Math. Res. Notices 1996, no. 13, 613--663.
  • T. Graber and R. Pandharipande, Localization of virtual classes, Invent. Math. 135 (1999), 487--518.
  • V. Guillemin, Kaehler structures on toric varieties, J. Differential Geom. 40 (1994), 285--309.
  • H. Hofer, On the topological properties of symplectic maps, Proc. Roy. Soc. Edinburgh Sect. A 115 (1990), 25--38.
  • K. Hori, ``Linear models of supersymmetric D-branes'' in Symplectic Geometry and Mirror Symmetry (Seoul, 2000), World Sci. Publishing, River Edge, New Jersey, 2001, 111--186.
  • K. Hori, S. Katz, A. Klemm, R. Pandharipande, R. Thomas, C. Vafa, R. Vakil, and E. Zaslow, Mirror Symmetry, Clay Math. Monogr. 1, Amer. Math. Soc., Providence, 2003.
  • K. Hori and C. Vafa, Mirror symmetry, preprint, 2000.
  • L. HöRmander, Fourier integral operators, I, Acta Math. 127 (1971), 79--183.
  • H. Iritani, Quantum D-modules and equivariant Floer theory for free loop spaces, Math. Z. 252 (2006), 577--622.
  • —, Convergence of quantum cohomology by quantum Lefschetz, J. Reine Angew. Math. 610 (2007), 29--69.
  • M. Kontsevich, ``Homological algebra of mirror symmetry'' in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), Birkhäuser, Basel, 1995.
  • A. G. Kouchnirenko [Kušnirenko], Polyèdres de Newton et nombres de Milnor, Invent. Math. 32 (1976), 1--31.
  • A.-M. Li and Y. Ruan, Symplectic surgery and Gromov-Witten invariants of Calabi-Yau 3-folds, Invent. Math. 145 (2001), 151--218.
  • Yu. I. Manin, Frobenius Manifolds, Quantum Cohomology, and Moduli Spaces, Amer. Math. Soc. Colloq. Publ. 47, Amer. Math. Soc., Providence, 1999.
  • J. Marsden and A. Weinstein, Reduction of Symplectic Manifolds with Symmetry, Rep. Mathematical Phys. 5 (1974), 121--130.
  • J. N. Mather, ``Stratifications and mappings'' in Dynamical Systems (Salvador, Brazil, 1971) Academic Press, New York, 1973, 195--232.
  • H. Matsumura, Commutative Algebra, Benjamin, New York, 1970.
  • D. Mcduff, Displacing Lagrangian toric fibers via probes, preprint.
  • D. Mcduff and S. Tolman, Topological properties of Hamiltonian circle actions, IMRP Int. Math. Res. Pap. 2006, no. 72826, 1--77.
  • J. Milnor, Singular Points of Complex Hypersurfaces, Ann. of Math. Stud. 61, Princeton Univ. Press, Princeton, 1968.
  • M. Nagata, Local rings, Interscience Tracts in Pure and Applied Mathematics 13, Interscience, New York, 1962.
  • Y.-G. Oh, Floer cohomology of Lagrangian intersections and pseudo-holomorphic discs, I, Comm. Pure Appl. Math. 46 (1993), 949--994.; Addenda, Comm. Pure Appl. Math. 48 (1995), 1299--1302.$\!$;
  • —, Symplectic topology as the geometry of action functional, II, Comm. Anal. Geom. 7 (1999), 1--55.
  • —, ``Construction of spectral invariants of Hamiltonian paths on closed symplectic manifolds'' in The Breadth of Symplectic and Poisson Geometry, Prog. Math. 232, Birkhäuser, Boston, 2005, 525--570.
  • —, Floer mini-max theory, the Cerf diagram, and the spectral invariants, J. Korean Math. Soc. 46 (2009), 363--447.
  • H. Ohta, ``Obstruction to and deformation of Lagrangian intersection Floer cohomology'' in Symplectic Geometry and Mirror Symmetry (Seoul, 2000), World Sci., River Edge, N.J., 2001, 281--309.
  • Y. Ruan, Virtual neighborhoods and pseudo-holomorphic curves, Turkish J. Math. 23 (1999), 161--231.
  • K. Saito, Period mapping associated to a primitive form, Publ. Res. Inst. Math. Sci. 19 (1983), 1231--1264.
  • M. Schwarz, On the action spectrum for closed symplectically aspherical manifolds, Pacific J. Math. 193 (2000), 419--461.
  • P. Seidel, $\pi\sb 1$ of symplectic automorphism groups and invertibles in quantum homology rings, Geom. Funct. Anal. 7 (1997), 1046--1095.
  • —, Fukaya Categories and Picard-Lefschetz Theory, Zur. Lect. Adv. Math., European Mathematical Society, Zürich, 2008.
  • K. Ueda, Homological mirror symmetry for toric del Pezzo surfaces, Comm. Math. Phys. 264 (2006), 71--85.
  • M. Usher, Spectral numbers in Floer theories, Compos. Math. 144 (2008), 1581--1592.
  • C. Viterbo, Symplectic topology as the geometry of generating functions, Math. Ann. 292 (1992), 685--710.