Abstract analogues of flux as symplectic invariants
Mémoires de la Société Mathématique de France, no. 137 (2014) , 141 p.
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We study families of objects in Fukaya categories, specifically ones whose deformation behaviour is prescribed by the choice of an odd degree cohomology class. This leads to invariants of symplectic manifolds, which we apply to blowups along symplectic mapping tori.

Nous étudions des familles d’objets dans des catégories de Fukaya, en particulier celles dont le comportement infinitésimal est determiné par une classe de cohomologie de degré impair. Cette étude aboutit à des invariants des variétés symplectiques ; nous en tirons des conséquences pour les éclatements de tores d’applications symplectiques.

DOI: 10.24033/msmf.447
Classification: 53D40,  16E45
Keywords: Fukaya categories, Flux homomorphism, Floer cohomology
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Seidel, Paul. Abstract analogues of flux as symplectic invariants. Mémoires de la Société Mathématique de France, Serie 2, , no. 137 (2014), 141 p. doi : 10.24033/msmf.447. http://numdam.org/item/MSMF_2014_2_137__1_0/

[1] V. G. A. Beilinson & W. Soergel« Koszul duality patterns in representation theory », J. Amer. Math. Soc. 9 (1996), p. 473–527. | Zbl | MR

[2] M. Abouzaid« A geometric criterion for generating the Fukaya category », Publ. Math. IHÉS 112 (2010), p. 191–240. | Numdam | Zbl | MR

[3] M. Abouzaid, K. Fukaya, Y.-G. Oh, H. Ohta & K. OnoIn preparation.

[4] M. Abouzaid & I. Smith« Homological mirror symmetry for the four-torus », Duke Math. J. 152 (2010), p. 373–440. | Zbl | MR

[5] P. Albers« A Lagrangian Piunikhin-Salamon-Schwarz morphism and two comparison homomorphisms in Floer homology », Int. Math. Res. Not. (2008), Art. ID 134, 56 pp. | Zbl | MR

[6] M. Atiyah« Complex analytic connections in fibre bundles », Trans. Amer. Math. Soc. 85 (1957), p. 181–207. | Zbl | MR

[7] D. Auroux« Mirror symmetry and T-duality in the complement of an anticanonical divisor », J. Gökova Geom. Topol. 1 (2007), p. 51–91. | Zbl | MR

[8] —, « Lecture notes from a topics course on mirror symmetry », (2009), available on the author’s webpage.

[9] A. Beilinson, V. Ginsburg & V. Schechtman« Koszul duality », J. Geom. Physics 5 (1988), p. 317–350. | Zbl | MR

[10] J. Bernstein« Algebraic theory of D-modules », unpublished lecture notes.

[11] P. Biran & O. Cornea« A Lagrangian quantum homology, New perspectives and challenges in symplectic field theory, pp. 1–44 », in CRM Proc. Lecture Notes, vol. 49, Amer. Math. Soc., 2009. | MR

[12] —, « Rigidity and uniruling for Lagrangian submanifolds », Geom. Topol. 13 (2009), p. 2881–2989. | Zbl | MR

[13] —, « Lagrangian cobordism. I », J. Amer. Math. Soc. 26 (2013), p. 295–340. | Zbl | MR

[14] A. Bondal & M. V. Den Bergh« Generators and representability of functors in commutative and noncommutative geometry », Moscow Math. J. 3 (2003), p. 1–36. | Zbl | MR

[15] A. Bondal & M. Kapranov« Enhanced triangulated categories », Math. USSR Sbornik 70 (1991), p. 93–107. | Zbl | MR

[16] F. Bourgeois« A Morse-Bott approach to contact homology », Thèse, Stanford University, 2002. | MR

[17] L. Buhovsky« The Maslov class of Lagrangian tori and quantum products in Floer cohomology », J. Topol. Anal. 2 (2010), p. 57–75. | Zbl | MR

[18] H. Cartan & S. EilenbergHomological Algebra, Princeton Univ. Press, 1956. | Zbl | MR

[19] C.-H. Cho & Y.-G. Oh« Floer cohomology and disc instantons of Lagrangian torus fibers in toric Fano manifolds », Asian J. Math. 10 (2006), p. 773–814. | Zbl | MR

[20] B. Conrad« Several approaches to non-Archimedean geometry, pp. 9–63 », in P-adic geometry, Univ. Lecture Ser., vol. 45, Amer. Math. Soc., 2008. | MR

[21] M. Datta« Immersions in a symplectic manifold », Proc. Indian Acad. Sci. Math. Sci. 108 (1998), p. 137–149. | Zbl | MR

[22] P. Deligne, P. Griffiths, J. Morgan & D. Sullivan« Real homotopy theory of Kähler manifolds », Invent. Math. 29 (1975), p. 245–274. | Zbl | EuDML | MR

[23] A. Dold« Zur Homotopietheorie der Kettenkomplexe », Math. Ann. 140 (1960), p. 278–298. | Zbl | EuDML | MR

[24] —, Lectures on Algebraic Topology, 2nd ed., Springer, 1980.

[25] S. Dostoglou & D. Salamon« Self dual instantons and holomorphic curves », Annals of Math. 139 (1994), p. 581–640. | Zbl | MR

[26] V. Drinfeld« DG quotients of DG categories », J. Algebra 272 (2004), p. 643–691. | Zbl | MR

[27] Y. Eliashberg & N. MishachevIntroduction to the h-principle, Amer. Math. Soc., 2002. | Zbl | MR

[28] A. Floer« Symplectic fixed points and holomorphic spheres », Commun. Math. Phys. 120 (1989), p. 575–611. | Zbl | MR

[29] R. Fröberg« Koszul algebras, pp. 337–350 », in Advances in commutative ring theory (Fez, 1997), Lecture Notes in Pure and Appl. Math., vol. 205, Dekker, 1999. | MR

[30] K. Fukaya« Cyclic symmetry and adic convergence in Lagrangian Floer theory », Preprint arXiv:0907.4219, 2009. | MR

[31] —, « Mirror symmetry of abelian varieties and multi-theta functions », J. Algebraic Geom. 11 (2002), p. 393–512. | Zbl | MR

[32] —, « Floer homology and mirror symmetry. II, pp. 31–127 », in Minimal surfaces, geometric analysis and symplectic geometry (Baltimore, MD, 1999), 2002.

[33] —, « Floer homology for families – a progress report, pp. 33–68 », in Integrable systems, topology, and physics (Tokyo, 2000), Amer. Math. Soc., 2002.

[34] —, « Floer homology of Lagrangian submanifolds », Sugaku Expositions 26 (2013), p. 99–127. | Zbl | MR

[35] K. Fukaya & Y.-G. Oh« Zero-loop open strings in the cotangent bundle and Morse homotopy », Asian J. Math. 1 (1998), p. 96–180. | Zbl | MR

[36] K. Fukaya, Y.-G. Oh, H. Ohta & K. Ono« Lagrangian surgery and holomorphic discs », preprint, 2007. Originally intended as Chapter 10 of [39]; this remains available from the first author’s homepage at Kyoto.

[37] —, « Canonical models of filtered A -algebras and Morse complexes, pp. 201–227 », in New perspectives and challenges in symplectic field theory, CRM Proc. Lecture Notes, vol. 49, Amer. Math. Soc., 2009. | MR

[38] —, « Lagrangian Floer theory on compact toric manifolds. I », Duke Math. J. 151 (2010), p. 23–174. | Zbl | MR

[39] —, « Lagrangian intersection Floer theory – anomaly and obstruction », Amer. Math. Soc. (2010).

[40] —, « Lagrangian Floer theory on compact toric manifolds. II: Bulk deformations », Selecta Math. 17 (2011), p. 609–711. | Zbl | MR

[41] S. Gitler« The cohomology of blow ups », Bol. Soc. Mat. Mex. 37 (1992), p. 167–175. | Zbl | MR

[42] W. Goldman & J. Millson« The deformation theory of the fundamental group of compact Kähler manifolds », IHÉS Publ. Math. 67 (1988), p. 43–96. | Numdam | Zbl | EuDML | MR

[43] I. Gordon« Symplectic reflection algebras, pp. 285-347 », in Trends in representation theory of algebras and related topics, Eur. Math. Soc., 2008. | MR

[44] E. Green, G. Hartman, E. Marcos & Ø. Solberg« Resolutions over Koszul algebras », Arch. Math. 85 (2005), p. 118–127. | Zbl | MR

[45] M. GromovPartial differential relations, Springer, 1986. | Zbl | MR

[46] J. Harer, A. Kas & R. KirbyHandlebody decompositions of complex surfaces, Mem. Amer. Math. Soc., vol. 62, Amer. Math. Soc., 1986. | Zbl | MR

[47] H. Hofer & D. Salamon« Floer homology and Novikov rings, pp. 483–524 », in The Floer memorial volume (H. Hofer, C. Taubes, A. Weinstein, and E. Zehnder, eds.), Progress in Mathematics, vol. 133, Birkhäuser, 1995. | Zbl | MR

[48] K. Hori & C. Vafa« Mirror symmetry », preprint hep-th/0002222, 2000.

[49] J. Hu, T.-J. Li & Y. Ruan« Birational cobordism invariance of uniruled symplectic manifolds », Invent. Math. 172 (2008), p. 231–275. | Zbl | MR

[50] D. Huybrechts & R. Thomas« Deformation-obstruction theory for complexes via Atiyah and Kodaira-Spencer classes », Math. Ann. 346 (2010), p. 545–569. | Zbl | MR

[51] E. Ionel & T. Parker« The symplectic sum formula for Gromov-Witten invariants », Ann. of Math. (2) 159 (2004), p. 935–1025. | Zbl | MR

[52] J. Johns« Complexifications of Morse functions and the directed Donaldson-Fukaya category », J. Symplectic Geom. 8 (2010), p. 403–500. | Zbl | MR

[53] T. Kadeishvili« The structure of the A -algebra, and the Hochschild and Harrison cohomologies (Russian) », Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR 91 (1988), p. 19–27. | MR

[54] B. Keller« Introduction to A-infinity algebras and modules », Homology Homotopy Appl. 3 (2001), p. 1–35. | Zbl | EuDML | MR

[55] —, « Hochschild cohomology and derived Picard groups », J. Pure Appl. Algebra 190 (2004), p. 177–196. | Zbl | MR

[56] —, « On differential graded categories, pp. 151–190 », in International Congress of Mathematicians. Vol. II, Eur. Math. Soc., 2006.

[57] M. Khovanov & P. Seidel« Quivers, Floer cohomology, and braid group actions », J. Amer. Math. Soc. 15 (2002), p. 203–271. | Zbl | MR

[58] M. Kontsevich« Homological algebra of mirror symmetry, pp. 120–139 », in Proceedings of the International Congress of Mathematicians (Zürich, 1994, Birkhäuser, 1995. | MR

[59] M. Kontsevich & Y. Soibelman« Homological mirror symmetry and torus fibrations, pp. 203–263 », in Symplectic geometry and mirror symmetry, World Scientific, 2001. | MR

[60] H. Krause« The stable derived category of a Noetherian scheme », Compos. Math. 141 (2005), p. 1128–1162. | Zbl | MR

[61] F. Lalonde, D. Mcduff & L. Polterovich« Topological rigidity of Hamiltonian loops and quantum homology », Invent. Math. 135 (1999), p. 369–385. | MR

[62] P. Lambrechts & D. Stanley« The rational homotopy type of a blow-up in the stable case », Geom. Topol. 12 (2008), p. 1921–1993. | Zbl | MR

[63] K. Lefevre« Sur les A -catégories », Thèse, Université Paris 7, 2002.

[64] Y. Lekili & M. Lipyanskiy« Geometric composition in quilted Floer theory », Adv. Math. 236 (2013), p. 1–23. | Zbl | MR

[65] Y. Lekili & T. Perutz« Fukaya categories of the torus and Dehn surgery », Proc. Natl. Acad. Sci. USA 108 (2011), p. 8106–8113. | Zbl | MR

[66] E. Lerman« Symplectic cuts », Math. Res. Lett. 2 (1995), p. 247–258. | Zbl | MR

[67] A.-M. Li & Y. Ruan« Symplectic surgery and Gromov-Witten invariants of Calabi-Yau 3-folds », Invent. Math. 145 (2001), p. 151–218. | Zbl | MR

[68] J.-L. LodayThe diagonal of the Stasheff polytope, Progress Math., vol. 287, Birkhäuser, 2011. | Zbl | MR

[69] N. Markarian« The Atiyah class, Hochschild cohomology and the Riemann-Roch theorem », J. London Math. Soc. 79 (2009), p. 129–143. | Zbl | MR

[70] S. Ma’U« Quilted Floer modules », Conference talk, recording available at http://media.scgp.stonybrook.edu/video/video.php?f=20110518_3_qtp.mp4, 2011.

[71] S. Ma’U, K. Wehrheim & C. Woodward« A-infinity functors for Lagrangian correspondences », Manuscript available on the second and third authors’ homepages.

[72] D. Maulik & R. Pandharipande« A topological view of Gromov-Witten theory », Topology 45 (2006), p. 887–918. | Zbl | MR

[73] D. Mcduff« Examples of symplectic structures », Invent. Math. 89 (1987), p. 13–36. | Zbl | EuDML | MR

[74] D. Mcduff & D. SalamonIntroduction to symplectic topology, 2nd ed., Oxford Univ. Press, 1998. | MR

[75] S. Mukai« Duality between D(X) and D(X ^) with its application to Picard sheaves », Nagoya J. Math. 81 (1981), p. 153–175. | Zbl | MR

[76] D. MumfordTata lectures on theta. I, Progress in Mathematics, vol. 28, Birkhäuser, 1983. | Zbl | MR

[77] Y.-G. Oh« Floer cohomology, spectral sequences, and the Maslov class of Lagrangian embeddings », Int. Math. Res. Notices (1996), p. 305–346. | Zbl | MR

[78] —, « Seidel’s long exact sequence on Calabi-Yau manifolds », Kyoto J. Math. 51 (2011), p. 687–765. | Zbl | MR

[79] Y.-G. Oh & K. Fukaya« Floer homology in symplectic geometry and in mirror symmetry, pp. 879-905 », in International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006. | MR

[80] K. Ono« Floer-Novikov cohomology and the flux conjecture », Geom. Funct. Anal. 16 (2006), p. 981–1020. | Zbl | MR

[81] M. Papikian« Rigid-analytic geometry and the uniformization of abelian varieties, pp. 145–160 », in Snowbird lectures in algebraic geometry, Contemp. Math., Amer. Math. Soc., 2005. | MR

[82] T. Perutz« Talk at the AIM workshop on Cyclic homology and symplectic topology », 2009.

[83] S. Piunikhin, D. Salamon & M. Schwarz« Symplectic Floer-Donaldson theory and quantum cohomology, pp. 171–200 », in Contact and symplectic geometry (C.B. Thomas, ed.), Cambridge Univ. Press, 1996. | Zbl | MR

[84] A. Polishchuk« A-infinity algebra of an elliptic curve and Eisenstein series », Comm. Math. Phys. 301 (2011), p. 709–722. | Zbl | MR

[85] A. Polishchuk & E. Zaslow« Categorical mirror symmetry: the elliptic curve », Adv. Theor. Math. Phys. 2 (1998), p. 443–470. | Zbl | MR

[86] M. PoźniakFloer homology, Novikov rings and clean intersections, Northern California Symplectic Geometry Seminar, Amer. Math. Soc., 1999. | Zbl | MR

[87] S. Priddy« Koszul resolutions », Trans. Amer. Math. Soc. 152 (1970), p. 39–60. | Zbl | MR

[88] M. Schwarz« A quantum cup-length estimate for symplectic fixed points », Invent. Math. 133 (1998), p. 353–397. | Zbl | MR

[89] P. SeidelHomological mirror symmetry for the quartic surface, Memoirs of the Amer. Math. Soc., to appear. | Zbl | MR

[90] —, « Floer homology and the symplectic isotopy problem », Thèse, Oxford Univ., 1997, Available on the author’s homepage.

[91] —, « Graded Lagrangian submanifolds », Bull. Soc. Math. France 128 (2000), p. 103–146. | Numdam | Zbl | EuDML

[92] —, « Braids and symplectic four-manifolds with abelian fundamental group, pp. 93–100 », in Gököva Geometry and Topology Conference Proceedings Special Volume, vol. 26, 2002. | MR

[93] —, « Fukaya categories and Picard-Lefschetz theory », European Math. Soc. (2008). | Zbl

[94] —, « Lectures on four-dimensional Dehn twists, pp. 231–267 », in Symplectic 4-manifolds and algebraic surfaces, Lecture Notes in Math., vol. 1938, Springer, 2008. | MR

[95] —, « Homological mirror symmetry for the genus two curve », J. Algebraic Geom. 20 (2011), p. 727–769. | Zbl | MR

[96] P. Seidel & J. Solomon« Symplectic cohomology and q-intersection numbers », Geom. Funct. Anal. 22 (2012), p. 443–477. | Zbl | MR

[97] P. Seidel & R. Thomas« Braid group actions on derived categories of coherent sheaves », Duke Math. J. 108 (2001), p. 37–108. | Zbl | MR

[98] N. Sheridan« On the Fukaya category of a Fano hypersurface in projective space », preprint ArXiv:1306.4143, 2013. | MR

[99] —, « On the homological mirror symmetry conjecture for pairs of pants », J. Diff. Geom. 89 (2011), p. 271–367. | Zbl | MR

[100] I. Smith« Floer cohomology and pencils of quadrics », Invent. Math. 189 (2012), p. 149–250. | Zbl | MR

[101] N. Spaltenstein« Resolutions of unbounded complexes », Compos. Math. 65 (1988), p. 121–154. | Numdam | Zbl | EuDML | MR

[102] C. Taubes« The Seiberg-Witten invariants and symplectic forms », Math. Research Letters 1 (1994), p. 809–822. | Zbl | MR

[103] B. Toën & M. Vaquié« Moduli of objects in dg-categories », Ann. Sci. École Norm. Sup. 40 (2007), p. 387–444. | Zbl | EuDML | MR

[104] K. Wehrheim & C. Woodward« Orientations for pseudoholomorphic quilts », manuscript, available on the first author’s homepage.

[105] —, « Functoriality for Lagrangian correspondences in Floer theory », Quantum Topol. 1 (2010), p. 129–170. | Zbl | MR

[106] —, « Quilted Floer cohomology », Geom. Topol. 14 (2010), p. 833–902. | Zbl | MR

[107] —, « Floer cohomology and geometric composition of Lagrangian correspondences », Adv. Math. 230 (2012), p. 177–228. | Zbl | MR

[108] E. Whittaker & G. WatsonA course of modern analysis, 4th ed., Cambridge Univ. Press, 1927. | JFM | MR

[109] A. Yekutieli« Dualizing complexes, Morita equivalence and the derived Picard group of a ring », J. London Math. Soc. (2) 60 (1999), p. 723–746. | Zbl | MR

[110] —, « The derived Picard group is a locally algebraic group », Algebr. Represent. Theory 7 (2004), p. 53–57. | Zbl | MR

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