Commuting symplectomorphisms and Dehn twists in divisors

Tonkonog, Dmitry¹

¹ Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK

Geometry & topology, Tome 19 (2015) no. 6, pp. 3345-3403.

Résumé

Two commuting symplectomorphisms of a symplectic manifold give rise to actions on Floer cohomologies of each other. We prove the elliptic relation saying that the supertraces of these two actions are equal. In the case when a symplectomorphism $f$ commutes with a symplectic involution, the elliptic relation provides a lower bound on the dimension of ${HF}^{\ast }\left(f\right)$ in terms of the Lefschetz number of $f$ restricted to the fixed locus of the involution. We apply this bound to prove that Dehn twists around vanishing Lagrangian spheres inside most hypersurfaces in Grassmannians have infinite order in the symplectic mapping class group.

Détail

DOI : 10.2140/gt.2015.19.3345

Classification : 53D40, 14F35, 14D05
Keywords: Floer cohomology, elliptic relation, symplectic involution, Dehn twist

Bibliographie

[1] M F Atiyah, On analytic surfaces with double points, Proc. Roy. Soc. London. Ser. A 247 (1958) 237

[2] R Avdek, Liouville hypersurfaces and connect sum cobordisms, preprint (2012)

[3] M C Beltrametti, M L Fania, A J Sommese, On the discriminant variety of a projective manifold, Forum Math. 4 (1992) 529

[4] D Ben-Zvi, D Nadler, Secondary traces, preprint (2013)

[5] M Betz, J Rade, Products and relations in symplectic Floer homology, preprint (2015)

[6] P Biran, C Membrez, The Lagrangian cubic equation, preprint (2015)

[7] F Bourgeois, Y Eliashberg, H Hofer, K Wysocki, E Zehnder, Compactness results in symplectic field theory, Geom. Topol. 7 (2003) 799

[8] F Bourgeois, K Mohnke, Coherent orientations in symplectic field theory, Math. Z. 248 (2004) 123

[9] M J Cowen, Automorphisms of Grassmannians, Proc. Amer. Math. Soc. 106 (1989) 99

[10] S Diaz, D Harbater, Strong Bertini theorems, Trans. Amer. Math. Soc. 324 (1991) 73

[11] S Dostoglou, D A Salamon, Self-dual instantons and holomorphic curves, Ann. of Math. 139 (1994) 581

[12] Y Eliashberg, A Givental, H Hofer, Introduction to symplectic field theory, Geom. Funct. Anal. (2000) 560

[13] A Floer, H Hofer, Coherent orientations for periodic orbit problems in symplectic geometry, Math. Z. 212 (1993) 13

[14] K Fukaya, Y G Oh, H Ohta, K Ono, Lagrangian intersection Floer theory: Anomaly and obstruction, I, AMS/IP Studies in Advanced Mathematics 46.1, Amer. Math. Soc. (2009)

[15] K Fukaya, K Ono, Arnold conjecture and Gromov–Witten invariant, Topology 38 (1999) 933

[16] H Hofer, D A Salamon, Floer homology and Novikov rings, from: "The Floer memorial volume" (editors H Hofer, C H Taubes, A Weinstein, E Zehnder), Progr. Math. 133, Birkhäuser (1995) 483

[17] T Holm, Hochschild cohomology rings of algebras $k[X]/(f)$, Beiträge Algebra Geom. 41 (2000) 291

[18] T V Kadeishvili, The structure of the $A(\infty)$–algebra, and the Hochschild and Harrison cohomologies, Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR 91 (1988) 19

[19] A M Keating, Dehn twists and free subgroups of symplectic mapping class groups, J. Topol. 7 (2014) 436

[20] M Khovanov, P Seidel, Quivers, Floer cohomology, and braid group actions, J. Amer. Math. Soc. 15 (2002) 203

[21] N A Krylov, Relative mapping class group of the trivial and the tangent disk bundles over the sphere, Pure Appl. Math. Q. 3 (2007) 631

[22] M Lönne, Fundamental groups of projective discriminant complements, Duke Math. J. 150 (2009) 357

[23] E Looijenga, Cohomology and intersection homology of algebraic varieties, from: "Complex algebraic geometry" (editor J Kollár), IAS/Park City Math. Ser. 3, Amer. Math. Soc. (1997) 221

[24] D Mcduff, D Salamon, $J$–holomorphic curves and symplectic topology, Amer. Math. Soc. Colloquium Publications 52, Amer. Math. Soc. (2004)

[25] S Mukai, Curves and Grassmannians, from: "Algebraic geometry and related topics" (editors J H Yang, Y Namikawa), Conf. Proc. Lecture Notes Algebraic Geom. 1, Int. Press (1993) 19

[26] Y G Oh, Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks, I, Comm. Pure Appl. Math. 46 (1993) 949

[27] Y G Oh, Seidel's long exact sequence on Calabi–Yau manifolds, Kyoto J. Math. 51 (2011) 687

[28] A Polishchuk, Lefschetz type formulas for dg-categories, Selecta Math. 20 (2014) 885

[29] A F Ritter, I Smith, The monotone wrapped Fukaya category and the open-closed string map, preprint (2015)

[30] D Salamon, E Zehnder, Morse theory for periodic solutions of Hamiltonian systems and the Maslov index, Comm. Pure Appl. Math. 45 (1992) 1303

[31] M Schwarz, Morse homology, Progress in Mathematics 111, Birkhäuser (1993)

[32] M Schwarz, Cohomology operations from $S^1$–cobordisms in Floer homology, PhD thesis, ETH Zürich (1995)

[33] P Seidel, Floer homology and the symplectic isotopy problem, PhD thesis, University of Oxford (1997)

[34] P Seidel, $\pi_1$ of symplectic automorphism groups and invertibles in quantum homology rings, Geom. Funct. Anal. 7 (1997) 1046

[35] P Seidel, Graded Lagrangian submanifolds, Bull. Soc. Math. France 128 (2000) 103

[36] P Seidel, A long exact sequence for symplectic Floer cohomology, Topology 42 (2003) 1003

[37] P Seidel, Fukaya categories and Picard–Lefschetz theory, European Math. Soc. (2008)

[38] P Seidel, Lectures on four-dimensional Dehn twists, from: "Symplectic 4–manifolds and algebraic surfaces" (editors F Catanese, G Tian), Lecture Notes in Math. 1938, Springer (2008) 231

[39] P Seidel, Abstract analogues of flux as symplectic invariants, Mém. Soc. Math. France 137, Soc. Math. France (2014) 135

[40] P Seidel, I Smith, Localization for involutions in Floer cohomology, Geom. Funct. Anal. 20 (2010) 1464

[41] N Sheridan, On the Fukaya category of a Fano hypersurface in projective space, preprint (2014)

[42] I Smith, Floer cohomology and pencils of quadrics, Invent. Math. 189 (2012) 149

[43] E A Tevelev, Invariant theory and algebraic transformation groups, IV: Projective duality and homogeneous spaces, Encyclopaedia of Mathematical Sciences 133, Springer (2005)

[44] K Wehrheim, C T Woodward, Functoriality for Lagrangian correspondences in Floer theory, Quantum Topol. 1 (2010) 129

[45] K Wehrheim, C T Woodward, Quilted Floer cohomology, Geom. Topol. 14 (2010) 833

[46] K Wehrheim, C Woodward, Exact triangle for fibered Dehn twists, preprint (2015)