Abstract analogues of flux as symplectic invariants  [ Analogues abstraits du flux comme invariants des variétés symplectiques ] (2014)


Seidel, Paul
Mémoires de la Société Mathématique de France, Tome 137 (2014) vi-135 p
Le texte intégral des articles récents est réservé aux abonnés de la revue. Consulter l'article sur le site de la revue
doi : 10.24033/msmf.447
URL stable : http://www.numdam.org/item?id=MSMF_2014_2_137__1_0

Bibliographie

[1] V. G. A. Beilinson & W. Soergel« Koszul duality patterns in representation theory », J. Amer. Math. Soc. 9 (1996), p. 473–527. Zbl 0864.17006 | MR 1322847

[2] M. Abouzaid« A geometric criterion for generating the Fukaya category », Publ. Math. IHÉS 112 (2010), p. 191–240. Numdam | Zbl 1215.53078 | MR 2737980

[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 1195.14056 | MR 2654219

[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 1158.53066 | MR 2424172

[6] M. Atiyah« Complex analytic connections in fibre bundles », Trans. Amer. Math. Soc. 85 (1957), p. 181–207. Zbl 0078.16002 | MR 86359

[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 1181.53076 | MR 2386535

[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 0695.14009 | MR 1048505

[10] J. Bernstein« Algebraic theory of D-modules », unpublished lecture notes.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 2555932

[12] —, « Rigidity and uniruling for Lagrangian submanifolds », Geom. Topol. 13 (2009), p. 2881–2989. Zbl 1180.53078 | MR 2546618

[13] —, « Lagrangian cobordism. I », J. Amer. Math. Soc. 26 (2013), p. 295–340. Zbl 1272.53071 | MR 3011416

[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 1135.18302 | MR 1996800

[15] A. Bondal & M. Kapranov« Enhanced triangulated categories », Math. USSR Sbornik 70 (1991), p. 93–107. Zbl 0729.18008 | MR 1055981

[16] F. Bourgeois« A Morse-Bott approach to contact homology », Thèse, Stanford University, 2002. MR 2703292

[17] L. Buhovsky« The Maslov class of Lagrangian tori and quantum products in Floer cohomology », J. Topol. Anal. 2 (2010), p. 57–75. Zbl 1235.53083 | MR 2646989

[18] H. Cartan & S. EilenbergHomological Algebra, Princeton Univ. Press, 1956. Zbl 0075.24305 | MR 77480

[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 1130.53055 | MR 2282365

[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 2482345

[21] M. Datta« Immersions in a symplectic manifold », Proc. Indian Acad. Sci. Math. Sci. 108 (1998), p. 137–149. Zbl 0957.53024 | MR 1631439

[22] P. Deligne, P. Griffiths, J. Morgan & D. Sullivan« Real homotopy theory of Kähler manifolds », Invent. Math. 29 (1975), p. 245–274. Zbl 0312.55011 | | MR 382702

[23] A. Dold« Zur Homotopietheorie der Kettenkomplexe », Math. Ann. 140 (1960), p. 278–298. Zbl 0093.36903 | | MR 112906

[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 0812.58031 | MR 1283871

[26] V. Drinfeld« DG quotients of DG categories », J. Algebra 272 (2004), p. 643–691. Zbl 1064.18009 | MR 2028075

[27] Y. Eliashberg & N. MishachevIntroduction to the h-principle, Amer. Math. Soc., 2002. Zbl 1008.58001 | MR 1909245

[28] A. Floer« Symplectic fixed points and holomorphic spheres », Commun. Math. Phys. 120 (1989), p. 575–611. Zbl 0755.58022 | MR 987770

[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 1767430

[30] K. Fukaya« Cyclic symmetry and adic convergence in Lagrangian Floer theory », Preprint arXiv:0907.4219, 2009. MR 2723862

[31] —, « Mirror symmetry of abelian varieties and multi-theta functions », J. Algebraic Geom. 11 (2002), p. 393–512. Zbl 1002.14014 | MR 1894935

[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 1305.53085 | MR 3089207

[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 0938.32009 | MR 1480992

[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 2555938

[38] —, « Lagrangian Floer theory on compact toric manifolds. I », Duke Math. J. 151 (2010), p. 23–174. Zbl 1190.53078 | MR 2573826

[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 1234.53023 | MR 2827178

[41] S. Gitler« The cohomology of blow ups », Bol. Soc. Mat. Mex. 37 (1992), p. 167–175. Zbl 0836.57018 | MR 1317571

[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 0678.53059 | | MR 972343

[43] I. Gordon« Symplectic reflection algebras, pp. 285-347 », in Trends in representation theory of algebras and related topics, Eur. Math. Soc., 2008. MR 2484729

[44] E. Green, G. Hartman, E. Marcos & Ø. Solberg« Resolutions over Koszul algebras », Arch. Math. 85 (2005), p. 118–127. Zbl 1096.16011 | MR 2161801

[45] M. GromovPartial differential relations, Springer, 1986. Zbl 0651.53001 | MR 864505

[46] J. Harer, A. Kas & R. KirbyHandlebody decompositions of complex surfaces, Mem. Amer. Math. Soc., vol. 62, Amer. Math. Soc., 1986. Zbl 0631.57009 | MR 849942

[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 0842.58029 | MR 1362838

[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 1163.53055 | MR 2390285

[50] D. Huybrechts & R. Thomas« Deformation-obstruction theory for complexes via Atiyah and Kodaira-Spencer classes », Math. Ann. 346 (2010), p. 545–569. Zbl 1186.14014 | MR 2578562

[51] E. Ionel & T. Parker« The symplectic sum formula for Gromov-Witten invariants », Ann. of Math. (2) 159 (2004), p. 935–1025. Zbl 1075.53092 | MR 2113018

[52] J. Johns« Complexifications of Morse functions and the directed Donaldson-Fukaya category », J. Symplectic Geom. 8 (2010), p. 403–500. Zbl 1211.53100 | MR 2738386

[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 1029003

[54] B. Keller« Introduction to A-infinity algebras and modules », Homology Homotopy Appl. 3 (2001), p. 1–35. Zbl 0989.18009 | | MR 1854636

[55] —, « Hochschild cohomology and derived Picard groups », J. Pure Appl. Algebra 190 (2004), p. 177–196. Zbl 1060.16010 | MR 2043327

[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 1035.53122 | MR 1862802

[58] M. Kontsevich« Homological algebra of mirror symmetry, pp. 120–139 », in Proceedings of the International Congress of Mathematicians (Zürich, 1994, Birkhäuser, 1995. MR 1403918

[59] M. Kontsevich & Y. Soibelman« Homological mirror symmetry and torus fibrations, pp. 203–263 », in Symplectic geometry and mirror symmetry, World Scientific, 2001. MR 1882331

[60] H. Krause« The stable derived category of a Noetherian scheme », Compos. Math. 141 (2005), p. 1128–1162. Zbl 1090.18006 | MR 2157133

[61] F. Lalonde, D. Mcduff & L. Polterovich« Topological rigidity of Hamiltonian loops and quantum homology », Invent. Math. 135 (1999), p. 369–385. MR 1666763

[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 1153.55010 | MR 2431012

[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 1268.53096 | MR 3019714

[65] Y. Lekili & T. Perutz« Fukaya categories of the torus and Dehn surgery », Proc. Natl. Acad. Sci. USA 108 (2011), p. 8106–8113. Zbl 1256.53055 | MR 2806646

[66] E. Lerman« Symplectic cuts », Math. Res. Lett. 2 (1995), p. 247–258. Zbl 0835.53034 | MR 1338784

[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 1062.53073 | MR 1839289

[68] J.-L. LodayThe diagonal of the Stasheff polytope, Progress Math., vol. 287, Birkhäuser, 2011. Zbl 1220.18007 | MR 2762549

[69] N. Markarian« The Atiyah class, Hochschild cohomology and the Riemann-Roch theorem », J. London Math. Soc. 79 (2009), p. 129–143. Zbl 1167.14005 | MR 2472137

[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 1112.14065 | MR 2248516

[73] D. Mcduff« Examples of symplectic structures », Invent. Math. 89 (1987), p. 13–36. Zbl 0625.53040 | | MR 892186

[74] D. Mcduff & D. SalamonIntroduction to symplectic topology, 2nd ed., Oxford Univ. Press, 1998. MR 1698616

[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 0417.14036 | MR 607081

[76] D. MumfordTata lectures on theta. I, Progress in Mathematics, vol. 28, Birkhäuser, 1983. Zbl 0744.14033 | MR 688651

[77] Y.-G. Oh« Floer cohomology, spectral sequences, and the Maslov class of Lagrangian embeddings », Int. Math. Res. Notices (1996), p. 305–346. Zbl 0858.58017 | MR 1389956

[78] —, « Seidel’s long exact sequence on Calabi-Yau manifolds », Kyoto J. Math. 51 (2011), p. 687–765. Zbl 1230.53079 | MR 2824005

[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 2275627

[80] K. Ono« Floer-Novikov cohomology and the flux conjecture », Geom. Funct. Anal. 16 (2006), p. 981–1020. Zbl 1113.53055 | MR 2276532

[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 2182895

[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 0874.53031 | MR 1432464

[84] A. Polishchuk« A-infinity algebra of an elliptic curve and Eisenstein series », Comm. Math. Phys. 301 (2011), p. 709–722. Zbl 1241.14008 | MR 2784277

[85] A. Polishchuk & E. Zaslow« Categorical mirror symmetry: the elliptic curve », Adv. Theor. Math. Phys. 2 (1998), p. 443–470. Zbl 0947.14017 | MR 1633036

[86] M. PoźniakFloer homology, Novikov rings and clean intersections, Northern California Symplectic Geometry Seminar, Amer. Math. Soc., 1999. Zbl 0948.57025 | MR 1736217

[87] S. Priddy« Koszul resolutions », Trans. Amer. Math. Soc. 152 (1970), p. 39–60. Zbl 0261.18016 | MR 265437

[88] M. Schwarz« A quantum cup-length estimate for symplectic fixed points », Invent. Math. 133 (1998), p. 353–397. Zbl 0951.53053 | MR 1632778

[89] P. SeidelHomological mirror symmetry for the quartic surface, Memoirs of the Amer. Math. Soc., to appear. Zbl 1334.53091 | MR 3364859

[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 0992.53059 |

[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 1892803

[93] —, « Fukaya categories and Picard-Lefschetz theory », European Math. Soc. (2008). Zbl 1159.53001

[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 2441414

[95] —, « Homological mirror symmetry for the genus two curve », J. Algebraic Geom. 20 (2011), p. 727–769. Zbl 1226.14028 | MR 2819674

[96] P. Seidel & J. Solomon« Symplectic cohomology and q-intersection numbers », Geom. Funct. Anal. 22 (2012), p. 443–477. Zbl 1250.53078 | MR 2929070

[97] P. Seidel & R. Thomas« Braid group actions on derived categories of coherent sheaves », Duke Math. J. 108 (2001), p. 37–108. Zbl 1092.14025 | MR 1831820

[98] N. Sheridan« On the Fukaya category of a Fano hypersurface in projective space », preprint ArXiv:1306.4143, 2013. MR 3578916

[99] —, « On the homological mirror symmetry conjecture for pairs of pants », J. Diff. Geom. 89 (2011), p. 271–367. Zbl 1255.53065 | MR 2863919

[100] I. Smith« Floer cohomology and pencils of quadrics », Invent. Math. 189 (2012), p. 149–250. Zbl 1255.14032 | MR 2929086

[101] N. Spaltenstein« Resolutions of unbounded complexes », Compos. Math. 65 (1988), p. 121–154. Numdam | Zbl 0636.18006 | | MR 932640

[102] C. Taubes« The Seiberg-Witten invariants and symplectic forms », Math. Research Letters 1 (1994), p. 809–822. Zbl 0853.57019 | MR 1306023

[103] B. Toën & M. Vaquié« Moduli of objects in dg-categories », Ann. Sci. École Norm. Sup. 40 (2007), p. 387–444. Zbl 1140.18005 | | MR 2493386

[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 1206.53088 | MR 2657646

[106] —, « Quilted Floer cohomology », Geom. Topol. 14 (2010), p. 833–902. Zbl 1205.53091 | MR 2602853

[107] —, « Floer cohomology and geometric composition of Lagrangian correspondences », Adv. Math. 230 (2012), p. 177–228. Zbl 1263.57026 | MR 2900542

[108] E. Whittaker & G. WatsonA course of modern analysis, 4th ed., Cambridge Univ. Press, 1927. Zbl 45.0433.02 | MR 1424469

[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 0954.16006 | MR 1753810

[110] —, « The derived Picard group is a locally algebraic group », Algebr. Represent. Theory 7 (2004), p. 53–57. Zbl 1075.18007 | MR 2046954