Generating the Fukaya categories of Hamiltonian 𝐺-manifolds

Evans, Jonathan¹ ; Lekili, Yankı²

¹ Department of Mathematics, University College London, London, United Kingdom
² Department of Mathematical Sciences, King’s College London, London, United Kingdom

Journal of the American Mathematical Society, Tome 32 (2019) no. 1, pp. 119-162

Résumé

Let $G$ be a compact Lie group, and let $k$ be a field of characteristic $p \geq 0$ such that $H^*(G)$ has no $p$-torsion if $p>0$. We show that a free Lagrangian orbit of a Hamiltonian $G$-action on a compact, monotone, symplectic manifold $X$ split-generates an idempotent summand of the monotone Fukaya category $\mathcal {F}(X; k)$ if and only if it represents a nonzero object of that summand (slightly more general results are also provided). Our result is based on an explicit understanding of the wrapped Fukaya category $\mathcal {W}(T^*G; k)$ through Koszul twisted complexes involving the zero-section and a cotangent fibre and on a functor $D^b \mathcal {W}(T^*G; k) \to D^b\mathcal {F}(X^{-} \times X; k)$ canonically associated to the Hamiltonian $G$-action on $X$. We explore several examples which can be studied in a uniform manner, including toric Fano varieties and certain Grassmannians.

Détail

DOI : 10.1090/jams/909

Bibliographie

[1] Abouzaid, Mohammed A geometric criterion for generating the Fukaya category Publ. Math. Inst. Hautes Études Sci. 2010 191 240

[2] Abouzaid, Mohammed A cotangent fibre generates the Fukaya category Adv. Math. 2011 894 939

[3] Abouzaid, Mohammed Nearby Lagrangians with vanishing Maslov class are homotopy equivalent Invent. Math. 2012 251 313

[4] Abouzaid, Mohammed On the wrapped Fukaya category and based loops J. Symplectic Geom. 2012 27 79

[5] Abouzaid, Mohammed, Seidel, Paul An open string analogue of Viterbo functoriality Geom. Topol. 2010 627 718

[6] Abouzaid, Mohammed, Smith, Ivan Homological mirror symmetry for the 4-torus Duke Math. J. 2010 373 440

[7] Adams, John Frank Infinite loop spaces 1978

[8] Atiyah, M. F., Macdonald, I. G. Introduction to commutative algebra 1969

[9] Avramov, Luchezar L., Iyengar, Srikanth Gaps in Hochschild cohomology imply smoothness for commutative algebras Math. Res. Lett. 2005 789 804

[10] Bedulli, Lucio, Gori, Anna Homogeneous Lagrangian submanifolds Comm. Anal. Geom. 2008 591 615

[11] Beilinson, Alexander, Ginzburg, Victor, Soergel, Wolfgang Koszul duality patterns in representation theory J. Amer. Math. Soc. 1996 473 527

[12] Biran, Paul, Cornea, Octav Rigidity and uniruling for Lagrangian submanifolds Geom. Topol. 2009 2881 2989

[13] Biran, Paul, Cornea, Octav Lagrangian topology and enumerative geometry Geom. Topol. 2012 963 1052

[14] Borel, Armand Sur l’homologie et la cohomologie des groupes de Lie compacts connexes Amer. J. Math. 1954 273 342

[15] Cho, Cheol-Hyun, Oh, Yong-Geun Floer cohomology and disc instantons of Lagrangian torus fibers in Fano toric manifolds Asian J. Math. 2006 773 814

[16] Clark, Allan Homotopy commutativity and the Moore spectral sequence Pacific J. Math. 1965 65 74

[17] Coates, Tom, Corti, Alessio, Galkin, Sergey, Golyshev, Vasily, Kasprzyk, Alexander Mirror symmetry and Fano manifolds 2013 285 300

[18] Damian, Mihai Floer homology on the universal cover, Audin’s conjecture and other constraints on Lagrangian submanifolds Comment. Math. Helv. 2012 433 462

[19] Dwyer, W. G. Strong convergence of the Eilenberg-Moore spectral sequence Topology 1974 255 265

[20] Dwyer, W. G., Greenlees, J. P. C. Complete modules and torsion modules Amer. J. Math. 2002 199 220

[21] Eilenberg, Samuel, Moore, John C. Homology and fibrations. I. Coalgebras, cotensor product and its derived functors Comment. Math. Helv. 1966 199 236

[22] Etgã¼, Tolga, Lekili, Yankä± Koszul duality patterns in Floer theory Geom. Topol. 2017 3313 3389

[23] Evans, Jonathan David, Lekili, Yankä± Floer cohomology of the Chiang Lagrangian Selecta Math. (N.S.) 2015 1361 1404

[24] Fukaya, Kenji, Oh, Yong-Geun, Ohta, Hiroshi, Ono, Kaoru Antisymplectic involution and Floer cohomology Geom. Topol. 2017 1 106

[25] Fukaya, Kenji, Oh, Yong-Geun, Ohta, Hiroshi, Ono, Kaoru Lagrangian intersection Floer theory: anomaly and obstruction. Part II 2009

[26] Fukaya, Kenji, Oh, Yong-Geun, Ohta, Hiroshi, Ono, Kaoru Lagrangian Floer theory on compact toric manifolds. I Duke Math. J. 2010 23 174

[27] Fukaya, Kenji, Seidel, Paul, Smith, Ivan Exact Lagrangian submanifolds in simply-connected cotangent bundles Invent. Math. 2008 1 27

[28] Fukaya, K., Seidel, P., Smith, I. The symplectic geometry of cotangent bundles from a categorical viewpoint 2009 1 26

[29] Gepner, Doron Fusion rings and geometry Comm. Math. Phys. 1991 381 411

[30] Guillemin, Victor, Sternberg, Shlomo The moment map revisited J. Differential Geom. 2005 137 162

[31] Haug, Luis On the quantum homology of real Lagrangians in Fano toric manifolds Int. Math. Res. Not. IMRN 2013 3171 3220

[32] Hopf, Heinz Über die Topologie der Gruppen-Mannigfaltigkeiten und ihre Verallgemeinerungen Ann. of Math. (2) 1941 22 52

[33] Iriyeh, Hiroshi Symplectic topology of Lagrangian submanifolds of ℂℙⁿ with intermediate minimal Maslov numbers Adv. Geom. 2017 247 264

[34] Iriyeh, Hiroshi, Sakai, Takashi, Tasaki, Hiroyuki Lagrangian Floer homology of a pair of real forms in Hermitian symmetric spaces of compact type J. Math. Soc. Japan 2013 1135 1151

[35] Kadeishvili, Tornike Cohomology 𝐶_{∞}-algebra and rational homotopy type 2009 225 240

[36] Kumpel, P. G., Jr. Lie groups and products of spheres Proc. Amer. Math. Soc. 1965 1350 1356

[37] Lekili, Yankä±, Lipyanskiy, Max Geometric composition in quilted Floer theory Adv. Math. 2013 1 23

[38] Mandell, Michael A. Cochains and homotopy type Publ. Math. Inst. Hautes Études Sci. 2006 213 246

[39] Mau, Sikimeti Luisa The multiplihedra in Lagrangian Floer theory 2008 180

[40] Ma’U, S., Wehrheim, K., Woodward, C. 𝐴_{∞} functors for Lagrangian correspondences Selecta Math. (N.S.) 2018 1913 2002

[41] Ma’U, S., Woodward, C. Geometric realizations of the multiplihedra Compos. Math. 2010 1002 1028

[42] Mcduff, Dusa, Salamon, Dietmar 𝐽-holomorphic curves and symplectic topology 2004

[43] Mimura, Mamoru Homotopy theory of Lie groups 1995 951 991

[44] Munkholm, Hans J. The Eilenberg-Moore spectral sequence and strongly homotopy multiplicative maps J. Pure Appl. Algebra 1974 1 50

[45] Nishinou, Takeo, Nohara, Yuichi, Ueda, Kazushi Potential functions via toric degenerations Proc. Japan Acad. Ser. A Math. Sci. 2012 31 33

[46] Nohara, Yuichi, Ueda, Kazushi Floer cohomologies of non-torus fibers of the Gelfand-Cetlin system J. Symplectic Geom. 2016 1251 1293

[47] Oh, Yong-Geun Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks. I Comm. Pure Appl. Math. 1993 949 993

[48] Oh, Yong-Geun Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks. II. (𝐶𝑃ⁿ,𝑅𝑃ⁿ) Comm. Pure Appl. Math. 1993 995 1012

[49] Oh, Y.-G. Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks. III. Arnol′d-Givental conjecture 1995 555 573

[50] Ostrover, Yaron, Tyomkin, Ilya On the quantum homology algebra of toric Fano manifolds Selecta Math. (N.S.) 2009 121 149

[51] Ritter, Alexander F. Circle actions, quantum cohomology, and the Fukaya category of Fano toric varieties Geom. Topol. 2016 1941 2052

[52] Ruan, Yongbin, Tian, Gang Bott-type symplectic Floer cohomology and its multiplication structures Math. Res. Lett. 1995 203 219

[53] Seidel, Paul Fukaya categories and Picard-Lefschetz theory 2008

[54] Seidel, Paul Homological mirror symmetry for the quartic surface Mem. Amer. Math. Soc. 2015

[55] Serre, Jean-Pierre Groupes d’homotopie et classes de groupes abéliens Ann. of Math. (2) 1953 258 294

[56] Sheridan, Nick On the Fukaya category of a Fano hypersurface in projective space Publ. Math. Inst. Hautes Études Sci. 2016 165 317

[57] Smith, Ivan Floer cohomology and pencils of quadrics Invent. Math. 2012 149 250

[58] Sugawara, Masahiro On the homotopy-commutativity of groups and loop spaces Mem. Coll. Sci. Univ. Kyoto Ser. A. Math. 1960/61 257 269

[59] Teleman, Constantin Gauge theory and mirror symmetry 2014 1309 1332

[60] Tonkonog, Dmitry The closed-open string map for 𝑆¹-invariant Lagrangians Algebr. Geom. Topol. 2018 15 68

[61] Wehrheim, Katrin, Woodward, Chris T. Pseudoholomorphic quilts J. Symplectic Geom. 2015 849 904

[62] Wehrheim, Katrin, Woodward, Chris T. Functoriality for Lagrangian correspondences in Floer theory Quantum Topol. 2010 129 170

[63] Wehrheim, Katrin, Woodward, Chris T. Floer cohomology and geometric composition of Lagrangian correspondences Adv. Math. 2012 177 228

[64] Wehrheim, Katrin, Woodward, Chris T. Quilted Floer trajectories with constant components: corrigendum to the article “Quilted Floer cohomology” [MR2602853] Geom. Topol. 2012 127 154

[65] Weinstein, Alan Symplectic geometry Bull. Amer. Math. Soc. (N.S.) 1981 1 13