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Floer cohomology and flips
About this Title
François Charest and Chris T. Woodward
Publication: Memoirs of the American Mathematical Society
Publication Year:
2022; Volume 279, Number 1372
ISBNs: 978-1-4704-5310-7 (print); 978-1-4704-7226-9 (online)
DOI: https://doi.org/10.1090/memo/1372
Published electronically: August 8, 2022
Table of Contents
Chapters
- 1. Introduction
- 2. Symplectic flips
- 3. Lagrangians associated to flips
- 4. Fukaya algebras
- 5. Homotopy invariance
- 6. Fukaya bimodules
- 7. Broken Fukaya algebras
- 8. The break-up process
Abstract
We show that blow-ups or reverse flips (in the sense of the minimal model program) of rational symplectic manifolds with point centers create Floer-non-trivial Lagrangian tori. These results are part of a conjectural decomposition of the Fukaya category of a compact symplectic manifold with a singularity-free running of the minimal model program, analogous to the description of Bondal-Orlov (Derived categories of coherent sheaves, 2002) and Kawamata (Derived categories of toric varieties, 2006) of the bounded derived category of coherent sheaves on a compact complex manifold.- Casim Abbas, An introduction to compactness results in symplectic field theory, Springer, Heidelberg, 2014. MR 3157146, DOI 10.1007/978-3-642-31543-5
- Mohammed Abouzaid, A geometric criterion for generating the Fukaya category, Publ. Math. Inst. Hautes Études Sci. 112 (2010), 191–240. MR 2737980, DOI 10.1007/s10240-010-0028-5
- Mohammed Abouzaid, Framed bordism and Lagrangian embeddings of exotic spheres, Ann. of Math. (2) 175 (2012), no. 1, 71–185. MR 2874640, DOI 10.4007/annals.2012.175.1.4
- Pedro Acosta and Mark Shoemaker, Quantum cohomology of toric blowups and Landau-Ginzburg correspondences, Algebr. Geom. 5 (2018), no. 2, 239–263. MR 3769893, DOI 10.14231/AG-2018-008
- S. Agnihotri and C. Woodward, Eigenvalues of products of unitary matrices and quantum Schubert calculus, Math. Res. Lett. 5 (1998), no. 6, 817–836. MR 1671192, DOI 10.4310/MRL.1998.v5.n6.a10
- M. F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 (1983), no. 1505, 523–615. MR 702806, DOI 10.1098/rsta.1983.0017
- Michèle Audin, The topology of torus actions on symplectic manifolds, Progress in Mathematics, vol. 93, Birkhäuser Verlag, Basel, 1991. Translated from the French by the author. MR 1106194, DOI 10.1007/978-3-0348-7221-8
- D. Auroux, Asymptotically holomorphic families of symplectic submanifolds, Geom. Funct. Anal. 7 (1997), no. 6, 971–995. MR 1487750, DOI 10.1007/s000390050033
- D. Auroux, A remark about Donaldson’s construction of symplectic submanifolds, J. Symplectic Geom. 1 (2002), no. 3, 647–658. MR 1959060, DOI 10.4310/JSG.2001.v1.n3.a4
- Denis Auroux, Damien Gayet, and Jean-Paul Mohsen, Symplectic hypersurfaces in the complement of an isotropic submanifold, Math. Ann. 321 (2001), no. 4, 739–754. MR 1872527, DOI 10.1007/s002080100248
- Arend Bayer, Semisimple quantum cohomology and blowups, Int. Math. Res. Not. 40 (2004), 2069–2083. MR 2064316, DOI 10.1155/S1073792804140907
- Caucher Birkar, Paolo Cascini, Christopher D. Hacon, and James McKernan, Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23 (2010), no. 2, 405–468. MR 2601039, DOI 10.1090/S0894-0347-09-00649-3
- I. Biswas and N. Raghavendra, Determinants of parabolic bundles on Riemann surfaces, Proc. Indian Acad. Sci. Math. Sci. 103 (1993), no. 1, 41–71. MR 1234199, DOI 10.1007/BF02837895
- Paul Biran and Octav Cornea, Rigidity and uniruling for Lagrangian submanifolds, Geom. Topol. 13 (2009), no. 5, 2881–2989. MR 2546618, DOI 10.2140/gt.2009.13.2881
- P. Biran and O. Cornea. Quantum structures for Lagrangian submanifolds. arxiv:0708.4221.
- J. M. Boardman and R. M. Vogt, Homotopy invariant algebraic structures on topological spaces, Lecture Notes in Mathematics, Vol. 347, Springer-Verlag, Berlin-New York, 1973. MR 420609
- D. Borthwick, T. Paul, and A. Uribe, Legendrian distributions with applications to relative Poincaré series, Invent. Math. 122 (1995), no. 2, 359–402. MR 1358981, DOI 10.1007/BF01231449
- Hans U. Boden and Yi Hu, Variations of moduli of parabolic bundles, Math. Ann. 301 (1995), no. 3, 539–559. MR 1324526, DOI 10.1007/BF01446645
- A. Bondal and D. Orlov, Derived categories of coherent sheaves, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002) Higher Ed. Press, Beijing, 2002, pp. 47–56. MR 1957019
- Lev A. Borisov, Linda Chen, and Gregory G. Smith, The orbifold Chow ring of toric Deligne-Mumford stacks, J. Amer. Math. Soc. 18 (2005), no. 1, 193–215. MR 2114820, DOI 10.1090/S0894-0347-04-00471-0
- Frédéric Bourgeois and Alexandru Oancea, Symplectic homology, autonomous Hamiltonians, and Morse-Bott moduli spaces, Duke Math. J. 146 (2009), no. 1, 71–174. MR 2475400, DOI 10.1215/00127094-2008-062
- F. Bourgeois, Y. Eliashberg, H. Hofer, K. Wysocki, and E. Zehnder, Compactness results in symplectic field theory, Geom. Topol. 7 (2003), 799–888. MR 2026549, DOI 10.2140/gt.2003.7.799
- Frédéric Bourgeois, A Morse-Bott approach to contact homology, Symplectic and contact topology: interactions and perspectives (Toronto, ON/Montreal, QC, 2001) Fields Inst. Commun., vol. 35, Amer. Math. Soc., Providence, RI, 2003, pp. 55–77. MR 1969267
- Michel Brion and Claudio Procesi, Action d’un tore dans une variété projective, Operator algebras, unitary representations, enveloping algebras, and invariant theory (Paris, 1989) Progr. Math., vol. 92, Birkhäuser Boston, Boston, MA, 1990, pp. 509–539 (French). MR 1103602, DOI 10.1007/s101070100288
- K. Cieliebak and K. Mohnke, Compactness for punctured holomorphic curves, J. Symplectic Geom. 3 (2005), no. 4, 589–654. Conference on Symplectic Topology. MR 2235856
- Francois Charest, Source Spaces and Perturbations for Cluster Complexes, ProQuest LLC, Ann Arbor, MI, 2012. Thesis (Ph.D.)–Universite de Montreal (Canada). MR 3153325
- François Charest and Chris Woodward, Floer trajectories and stabilizing divisors, J. Fixed Point Theory Appl. 19 (2017), no. 2, 1165–1236. MR 3659006, DOI 10.1007/s11784-016-0379-8
- Kai Cieliebak and Klaus Mohnke, Symplectic hypersurfaces and transversality in Gromov-Witten theory, J. Symplectic Geom. 5 (2007), no. 3, 281–356. MR 2399678
- Tom Coates, Hiroshi Iritani, Yunfeng Jiang, and Ed Segal, $K$-theoretic and categorical properties of toric Deligne-Mumford stacks, Pure Appl. Math. Q. 11 (2015), no. 2, 239–266. MR 3544765, DOI 10.4310/PAMQ.2015.v11.n2.a3
- Cheol-Hyun Cho, Products of Floer cohomology of torus fibers in toric Fano manifolds, Comm. Math. Phys. 260 (2005), no. 3, 613–640. MR 2183959, DOI 10.1007/s00220-005-1421-7
- Cheol-Hyun Cho and Yong-Geun Oh, Floer cohomology and disc instantons of Lagrangian torus fibers in Fano toric manifolds, Asian J. Math. 10 (2006), no. 4, 773–814. MR 2282365, DOI 10.4310/AJM.2006.v10.n4.a10
- Octav Cornea and François Lalonde, Cluster homology: an overview of the construction and results, Electron. Res. Announc. Amer. Math. Soc. 12 (2006), 1–12. MR 2200949, DOI 10.1090/S1079-6762-06-00154-5
- David A. Cox, John B. Little, and Henry K. Schenck, Toric varieties, Graduate Studies in Mathematics, vol. 124, American Mathematical Society, Providence, RI, 2011. MR 2810322, DOI 10.1090/gsm/124
- Thomas Delzant, Hamiltoniens périodiques et images convexes de l’application moment, Bull. Soc. Math. France 116 (1988), no. 3, 315–339 (French, with English summary). MR 984900
- S. K. Donaldson, Symplectic submanifolds and almost-complex geometry, J. Differential Geom. 44 (1996), no. 4, 666–705. MR 1438190
- J.-M. Drezet and M. S. Narasimhan, Groupe de Picard des variétés de modules de fibrés semi-stables sur les courbes algébriques, Invent. Math. 97 (1989), no. 1, 53–94 (French). MR 999313, DOI 10.1007/BF01850655
- J. J. Duistermaat and G. J. Heckman, Addendum to: “On the variation in the cohomology of the symplectic form of the reduced phase space”, Invent. Math. 72 (1983), no. 1, 153–158. MR 696693, DOI 10.1007/BF01389132
- Igor V. Dolgachev and Yi Hu, Variation of geometric invariant theory quotients, Inst. Hautes Études Sci. Publ. Math. 87 (1998), 5–56. With an appendix by Nicolas Ressayre. MR 1659282
- Y. Eliashberg, A. Givental, and H. Hofer, Introduction to symplectic field theory, Geom. Funct. Anal. Special Volume (2000), 560–673. GAFA 2000 (Tel Aviv, 1999). MR 1826267, DOI 10.1007/978-3-0346-0425-3_{4}
- Andreas Floer, Morse theory for Lagrangian intersections, J. Differential Geom. 28 (1988), no. 3, 513–547. MR 965228
- A. Floer, Monopoles on asymptotically flat manifolds, The Floer memorial volume, Progr. Math., vol. 133, Birkhäuser, Basel, 1995, pp. 3–41. MR 1362821
- Andreas Floer, Helmut Hofer, and Dietmar Salamon, Transversality in elliptic Morse theory for the symplectic action, Duke Math. J. 80 (1995), no. 1, 251–292. MR 1360618, DOI 10.1215/S0012-7094-95-08010-7
- K. Fukaya. Floer homology for $3$-manifolds with boundary I, 1999. unpublished manuscript.
- Urs Frauenfelder and Kai Zehmisch, Gromov compactness for holomorphic discs with totally real boundary conditions, J. Fixed Point Theory Appl. 17 (2015), no. 3, 521–540. MR 3411803, DOI 10.1007/s11784-015-0229-0
- Kenji Fukaya, Morse homotopy, $A^\infty$-category, and Floer homologies, Proceedings of GARC Workshop on Geometry and Topology ’93 (Seoul, 1993) Lecture Notes Ser., vol. 18, Seoul Nat. Univ., Seoul, 1993, pp. 1–102. MR 1270931
- Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono, Lagrangian intersection Floer theory: anomaly and obstruction. Part I, AMS/IP Studies in Advanced Mathematics, vol. 46, American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2009. MR 2553465, DOI 10.1090/amsip/046.1
- Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono, Lagrangian Floer theory on compact toric manifolds. I, Duke Math. J. 151 (2010), no. 1, 23–174. MR 2573826, DOI 10.1215/00127094-2009-062
- Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono, Lagrangian Floer theory on compact toric manifolds II: bulk deformations, Selecta Math. (N.S.) 17 (2011), no. 3, 609–711. MR 2827178, DOI 10.1007/s00029-011-0057-z
- K. Fukaya, Y.-G. Oh, H. Ohta, and K. Ono. Anti-symplectic involution and Floer cohomology. Geom. Topol. 21 (2017), no. 1, 1–106. arxiv:0912.2646.
- S. Ganatra. Symplectic Cohomology and Duality for the Wrapped Fukaya Category. PhD Thesis, Massachusetts Institute of Technology, 2006.
- William M. Goldman, Invariant functions on Lie groups and Hamiltonian flows of surface group representations, Invent. Math. 85 (1986), no. 2, 263–302. MR 846929, DOI 10.1007/BF01389091
- Robert E. Gompf, A new construction of symplectic manifolds, Ann. of Math. (2) 142 (1995), no. 3, 527–595. MR 1356781, DOI 10.2307/2118554
- Eduardo González and Chris T. Woodward, Quantum cohomology and toric minimal model programs, Adv. Math. 353 (2019), 591–646. MR 3986375, DOI 10.1016/j.aim.2019.07.004
- Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1994. Reprint of the 1978 original. MR 1288523, DOI 10.1002/9781118032527
- V. Guillemin and S. Sternberg, Birational equivalence in the symplectic category, Invent. Math. 97 (1989), no. 3, 485–522. MR 1005004, DOI 10.1007/BF01388888
- Victor Guillemin, Eugene Lerman, and Shlomo Sternberg, Symplectic fibrations and multiplicity diagrams, Cambridge University Press, Cambridge, 1996. MR 1414677, DOI 10.1017/CBO9780511574788
- Victor Guillemin and Shlomo Sternberg, Symplectic techniques in physics, 2nd ed., Cambridge University Press, Cambridge, 1990. MR 1066693
- Christopher D. Hacon and James McKernan, Flips and flops, Proceedings of the International Congress of Mathematicians. Volume II, Hindustan Book Agency, New Delhi, 2010, pp. 513–539. MR 2827807
- Christopher D. Hacon and James McKernan, The Sarkisov program, J. Algebraic Geom. 22 (2013), no. 2, 389–405. MR 3019454, DOI 10.1090/S1056-3911-2012-00599-2
- Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 463157
- Helmut Hofer, Kris Wysocki, and Eduard Zehnder, sc-smoothness, retractions and new models for smooth spaces, Discrete Contin. Dyn. Syst. 28 (2010), no. 2, 665–788. MR 2644764, DOI 10.3934/dcds.2010.28.665
- K. Hori and C. Vafa. Mirror symmetry. arxiv:hep-th/0002222
- Michael Hutchings and Clifford Henry Taubes, Gluing pseudoholomorphic curves along branched covered cylinders. II, J. Symplectic Geom. 7 (2009), no. 1, 29–133. MR 2491716
- Eleny-Nicoleta Ionel and Thomas H. Parker, Relative Gromov-Witten invariants, Ann. of Math. (2) 157 (2003), no. 1, 45–96. MR 1954264, DOI 10.4007/annals.2003.157.45
- L. C. Jeffrey and J. Weitsman, Toric structures on the moduli space of flat connections on a Riemann surface: volumes and the moment map, Adv. Math. 106 (1994), no. 2, 151–168. MR 1279216, DOI 10.1006/aima.1994.1054
- Yujiro Kawamata, Derived categories of toric varieties, Michigan Math. J. 54 (2006), no. 3, 517–535. MR 2280493, DOI 10.1307/mmj/1163789913
- George Kempf and Linda Ness, The length of vectors in representation spaces, Algebraic geometry (Proc. Summer Meeting, Univ. Copenhagen, Copenhagen, 1978) Lecture Notes in Math., vol. 732, Springer, Berlin, 1979, pp. 233–243. MR 555701
- Frances Clare Kirwan, Cohomology of quotients in symplectic and algebraic geometry, Mathematical Notes, vol. 31, Princeton University Press, Princeton, NJ, 1984. MR 766741, DOI 10.2307/j.ctv10vm2m8
- Steven L. Kleiman, Toward a numerical theory of ampleness, Ann. of Math. (2) 84 (1966), 293–344. MR 206009, DOI 10.2307/1970447
- Alexander A. Klyachko, Spatial polygons and stable configurations of points in the projective line, Algebraic geometry and its applications (Yaroslavl′, 1992) Aspects Math., E25, Friedr. Vieweg, Braunschweig, 1994, pp. 67–84. MR 1282021
- János Kollár and Shigefumi Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998. With the collaboration of C. H. Clemens and A. Corti; Translated from the 1998 Japanese original. MR 1658959, DOI 10.1017/CBO9780511662560
- Maxim Kontsevich, Homological algebra of mirror symmetry, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) Birkhäuser, Basel, 1995, pp. 120–139. MR 1403918
- M. Kontsevich and Yu. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys. 164 (1994), no. 3, 525–562. MR 1291244
- M. Kontsevich and Y. Soibelman, Notes on $A_\infty$-algebras, $A_\infty$-categories and non-commutative geometry, Homological mirror symmetry, Lecture Notes in Phys., vol. 757, Springer, Berlin, 2009, pp. 153–219. MR 2596638
- Y. Iwao, Y.-P. Lee, H.-W. Lin, and C.-L. Wang, Invariance of Gromov-Witten theory under a simple flop, J. Reine Angew. Math. 663 (2012), 67–90. MR 2889709, DOI 10.1515/CRELLE.2011.097
- Yuan-Pin Lee, Hui-Wen Lin, and Chin-Lung Wang, Flops, motives, and invariance of quantum rings, Ann. of Math. (2) 172 (2010), no. 1, 243–290. MR 2680420, DOI 10.4007/annals.2010.172.243
- K. Lefèvre-Hasegawa. Sur les $A_\infty$-catégories. PhD thesis, Université Paris 7, 2003.
- Eugene Lerman, Symplectic cuts, Math. Res. Lett. 2 (1995), no. 3, 247–258. MR 1338784, DOI 10.4310/MRL.1995.v2.n3.a2
- Eugene Lerman and Susan Tolman, Hamiltonian torus actions on symplectic orbifolds and toric varieties, Trans. Amer. Math. Soc. 349 (1997), no. 10, 4201–4230. MR 1401525, DOI 10.1090/S0002-9947-97-01821-7
- Yin Li, Disjoinable Lagrangian tori and semisimple symplectic cohomology, Algebr. Geom. Topol. 20 (2020), no. 5, 2269–2335. MR 4171567, DOI 10.2140/agt.2020.20.2269
- Jun Li, A degeneration formula of GW-invariants, J. Differential Geom. 60 (2002), no. 2, 199–293. MR 1938113
- Robert B. Lockhart and Robert C. McOwen, Elliptic differential operators on noncompact manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 12 (1985), no. 3, 409–447. MR 837256
- Charles-Michel Marle, Sous-variétés de rang constant d’une variété symplectique, Third Schnepfenried geometry conference, Vol. 1 (Schnepfenried, 1982) Astérisque, vol. 107, Soc. Math. France, Paris, 1983, pp. 69–86 (French). MR 753130
- E. Meinrenken and C. Woodward, Hamiltonian loop group actions and Verlinde factorization, J. Differential Geom. 50 (1998), no. 3, 417–469. MR 1690736
- E. Meinrenken and C. Woodward, Canonical bundles for Hamiltonian loop group manifolds, Pacific J. Math. 198 (2001), no. 2, 477–487. MR 1835519, DOI 10.2140/pjm.2001.198.477
- John Milnor, Lectures on the $h$-cobordism theorem, Princeton University Press, Princeton, NJ, 1965. Notes by L. Siebenmann and J. Sondow. MR 190942
- An-Min Li and Yongbin Ruan, Symplectic surgery and Gromov-Witten invariants of Calabi-Yau 3-folds, Invent. Math. 145 (2001), no. 1, 151–218. MR 1839289, DOI 10.1007/s002220100146
- J. D. Lotay and T. Pacini. Coupled flows, convexity and calibrations: Lagrangian and totally real geometry. 1404.4227.
- Emmanuel Opshtein, Singular polarizations and ellipsoid packings, Int. Math. Res. Not. IMRN 11 (2013), 2568–2600. MR 3065088, DOI 10.1093/imrn/rns137
- Tommaso Pacini, Maslov, Chern-Weil and mean curvature, J. Geom. Phys. 135 (2019), 129–134. MR 3872628, DOI 10.1016/j.geomphys.2018.09.009
- S. Mau, K. Wehrheim, and C.T. Woodward. ${A}_\infty$-functors for Lagrangian correspondences. in preparation.
- S. Ma’u and C. Woodward, Geometric realizations of the multiplihedra, Compos. Math. 146 (2010), no. 4, 1002–1028. MR 2660682, DOI 10.1112/S0010437X0900462X
- Dusa McDuff, Examples of simply-connected symplectic non-Kählerian manifolds, J. Differential Geom. 20 (1984), no. 1, 267–277. MR 772133
- Dusa McDuff, Displacing Lagrangian toric fibers via probes, Low-dimensional and symplectic topology, Proc. Sympos. Pure Math., vol. 82, Amer. Math. Soc., Providence, RI, 2011, pp. 131–160. MR 2768658, DOI 10.1090/pspum/082/2768658
- Dusa McDuff and Dietmar Salamon, $J$-holomorphic curves and symplectic topology, American Mathematical Society Colloquium Publications, vol. 52, American Mathematical Society, Providence, RI, 2004. MR 2045629, DOI 10.1090/coll/052
- V. B. Mehta and C. S. Seshadri, Moduli of vector bundles on curves with parabolic structures, Math. Ann. 248 (1980), no. 3, 205–239. MR 575939, DOI 10.1007/BF01420526
- Eckhard Meinrenken, Symplectic surgery and the $\textrm {Spin}^c$-Dirac operator, Adv. Math. 134 (1998), no. 2, 240–277. MR 1617809, DOI 10.1006/aima.1997.1701
- Michael Kapovich and John J. Millson, The symplectic geometry of polygons in Euclidean space, J. Differential Geom. 44 (1996), no. 3, 479–513. MR 1431002
- Han-Bom Moon and Sang-Bum Yoo, Birational geometry of the moduli space of rank 2 parabolic vector bundles on a rational curve, Int. Math. Res. Not. IMRN 3 (2016), 827–859. MR 3493435, DOI 10.1093/imrn/rnv154
- D. Mumford, J. Fogarty, and F. Kirwan, Geometric invariant theory, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34, Springer-Verlag, Berlin, 1994. MR 1304906
- Takeo Nishinou, Yuichi Nohara, and Kazushi Ueda, Toric degenerations of Gelfand-Cetlin systems and potential functions, Adv. Math. 224 (2010), no. 2, 648–706. MR 2609019, DOI 10.1016/j.aim.2009.12.012
- Yuichi Nohara and Kazushi Ueda, Toric degenerations of integrable systems on Grassmannians and polygon spaces, Nagoya Math. J. 214 (2014), 125–168. MR 3211821, DOI 10.1215/00277630-2643839
- Boris Pasquier, An approach of the minimal model program for horospherical varieties via moment polytopes, J. Reine Angew. Math. 708 (2015), 173–212. MR 3420333, DOI 10.1515/crelle-2013-0103
- Yong-Geun Oh, Floer cohomology, spectral sequences, and the Maslov class of Lagrangian embeddings, Internat. Math. Res. Notices 7 (1996), 305–346. MR 1389956, DOI 10.1155/S1073792896000219
- Yong-Geun Oh, Riemann-Hilbert problem and application to the perturbation theory of analytic discs, Kyungpook Math. J. 35 (1995), no. 1, 39–75. MR 1345070
- Yong-Geun Oh, Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks. I, Comm. Pure Appl. Math. 46 (1993), no. 7, 949–993. MR 1223659, DOI 10.1002/cpa.3160460702
- J. Palmer and C. Woodward. Immersed Floer cohomology and mean curvature flow. arXiv:1804.06799.
- Brett Parker, Holomorphic curves in exploded manifolds: compactness, Adv. Math. 283 (2015), 377–457. MR 3383807, DOI 10.1016/j.aim.2015.07.011
- Marcin Poźniak, Floer homology, Novikov rings and clean intersections, Northern California Symplectic Geometry Seminar, Amer. Math. Soc. Transl. Ser. 2, vol. 196, Amer. Math. Soc., Providence, RI, 1999, pp. 119–181. MR 1736217, DOI 10.1090/trans2/196/08
- Miles Reid, Decomposition of toric morphisms, Arithmetic and geometry, Vol. II, Progr. Math., vol. 36, Birkhäuser Boston, Boston, MA, 1983, pp. 395–418. MR 717617
- M. Reid. What is a flip? Notes from a Utah seminar 1982, available at http://homepages.warwick.ac.uk/$\sim$masda/3folds/.
- Yongbin Ruan, Surgery, quantum cohomology and birational geometry, Northern California Symplectic Geometry Seminar, Amer. Math. Soc. Transl. Ser. 2, vol. 196, Amer. Math. Soc., Providence, RI, 1999, pp. 183–198. MR 1736218, DOI 10.1090/trans2/196/09
- F. Schmäschke. Floer homology of Lagrangians in clean intersection. arXiv:1606.05327
- M. Schwarz. Cohomology Operations from $S^1$-Cobordisms in Floer Homology. PhD thesis, ETH Zurich, 1995
- Matthias Schwarz, Morse homology, Progress in Mathematics, vol. 111, Birkhäuser Verlag, Basel, 1993. MR 1239174, DOI 10.1007/978-3-0348-8577-5
- Paul Seidel, Graded Lagrangian submanifolds, Bull. Soc. Math. France 128 (2000), no. 1, 103–149 (English, with English and French summaries). MR 1765826
- Paul Seidel, Homological mirror symmetry for the genus two curve, J. Algebraic Geom. 20 (2011), no. 4, 727–769. MR 2819674, DOI 10.1090/S1056-3911-10-00550-3
- Paul Seidel, Homological mirror symmetry for the quartic surface, Mem. Amer. Math. Soc. 236 (2015), no. 1116, vi+129. MR 3364859, DOI 10.1090/memo/1116
- Paul Seidel, $A_\infty$-subalgebras and natural transformations, Homology Homotopy Appl. 10 (2008), no. 2, 83–114. MR 2426130
- Paul Seidel, Fukaya categories and Picard-Lefschetz theory, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2008. MR 2441780, DOI 10.4171/063
- Paul Seidel, Suspending Lefschetz fibrations, with an application to local mirror symmetry, Comm. Math. Phys. 297 (2010), no. 2, 515–528. MR 2651908, DOI 10.1007/s00220-009-0944-8
- Nick Sheridan, On the Fukaya category of a Fano hypersurface in projective space, Publ. Math. Inst. Hautes Études Sci. 124 (2016), 165–317. MR 3578916, DOI 10.1007/s10240-016-0082-8
- Jean-Claude Sikorav, Some properties of holomorphic curves in almost complex manifolds, Holomorphic curves in symplectic geometry, Progr. Math., vol. 117, Birkhäuser, Basel, 1994, pp. 165–189. MR 1274929, DOI 10.1007/978-3-0348-8508-9_{6}
- K. Smoczyk. Lagrangian mean curvature flow. Habilitation Thesis, Leipzig, 2001. http://service.ifam.uni-hannover.de/ smoczyk/publications/preprint07.pdf.
- Jian Song and Gang Tian, The Kähler-Ricci flow on surfaces of positive Kodaira dimension, Invent. Math. 170 (2007), no. 3, 609–653. MR 2357504, DOI 10.1007/s00222-007-0076-8
- Ivan Smith, Floer cohomology and pencils of quadrics, Invent. Math. 189 (2012), no. 1, 149–250. MR 2929086, DOI 10.1007/s00222-011-0364-1
- J.-M. Souriau, Structure des systèmes dynamiques, Dunod, Paris, 1970 (French). Maîtrises de mathématiques. MR 260238
- James Stasheff, $H$-spaces from a homotopy point of view, Lecture Notes in Mathematics, Vol. 161, Springer-Verlag, Berlin-New York, 1970. MR 270372
- M. Tehrani and A. Zinger. On Symplectic Sum Formulas in Gromov-Witten Theory. arXiv:1404.1898
- Thomas Treloar, The symplectic geometry of polygons in the 3-sphere, Canad. J. Math. 54 (2002), no. 1, 30–54. MR 1880958, DOI 10.4153/CJM-2002-002-1
- Michael Thaddeus, Geometric invariant theory and flips, J. Amer. Math. Soc. 9 (1996), no. 3, 691–723. MR 1333296, DOI 10.1090/S0894-0347-96-00204-4
- S. Venugopolan, C. Woodward, and G. Xu. Fukaya categories of blowups. 71 pages. arXiv:2006.12264.
- K. Wehrheim and C.T. Woodward. Orientations for pseudoholomorphic quilts. arXiv:1503.07803.
- Christopher T. Woodward, Gauged Floer theory of toric moment fibers, Geom. Funct. Anal. 21 (2011), no. 3, 680–749. MR 2810861, DOI 10.1007/s00039-011-0119-6
- G. Xu and C. T. Woodward. Partly-local domain-dependent almost complex structures. arXiv:1903.05557.