Disjoinable Lagrangian tori and semisimple symplectic cohomology

Li, Yin¹

¹Department of Mathematics, King’s College London, London, United Kingdom

Algebraic and Geometric Topology, Tome 20 (2020) no. 5, pp. 2269-2335

Résumé

We derive constraints on Lagrangian embeddings in completions of certain stable symplectic fillings whose symplectic cohomologies are semisimple. Manifolds with these properties can be constructed by generalizing the boundary connected sum operation to our setting, and are related to birational surgeries like blow-downs and flips. As a consequence, there are many nontoric (noncompact) monotone symplectic manifolds whose wrapped Fukaya categories are proper.

Détail

DOI : 10.2140/agt.2020.20.2269

Classification : 53D05, 53D35, 53D37, 53D40, 53D45
Keywords: symplectic cohomology, symplectic filling, flip

Bibliographie

[1] M Abouzaid, A geometric criterion for generating the Fukaya category, Publ. Math. Inst. Hautes Études Sci. 112 (2010) 191 | DOI

[2] M Abouzaid, D Auroux, L Katzarkov, Lagrangian fibrations on blowups of toric varieties and mirror symmetry for hypersurfaces, Publ. Math. Inst. Hautes Études Sci. 123 (2016) 199 | DOI

[3] M Abouzaid, P Seidel, An open string analogue of Viterbo functoriality, Geom. Topol. 14 (2010) 627 | DOI

[4] P Albers, J Kang, Vanishing of Rabinowitz Floer homology on negative line bundles, Math. Z. 285 (2017) 493 | DOI

[5] P Albers, M Mclean, Non-displaceable contact embeddings and infinitely many leaf-wise intersections, J. Symplectic Geom. 9 (2011) 271 | DOI

[6] D Auroux, L Katzarkov, D Orlov, Mirror symmetry for del Pezzo surfaces : vanishing cycles and coherent sheaves, Invent. Math. 166 (2006) 537 | DOI

[7] P Biran, O Cornea, Quantum structures for Lagrangian submanifolds, preprint (2007)

[8] P Biran, O Cornea, Rigidity and uniruling for Lagrangian submanifolds, Geom. Topol. 13 (2009) 2881 | DOI

[9] W M Boothby, H C Wang, On contact manifolds, Ann. of Math. 68 (1958) 721 | DOI

[10] F Charest, C T Woodward, Floer cohomology and flips, preprint (2015)

[11] C H Cho, Products of Floer cohomology of torus fibers in toric Fano manifolds, Comm. Math. Phys. 260 (2005) 613 | DOI

[12] K Cieliebak, Handle attaching in symplectic homology and the chord conjecture, J. Eur. Math. Soc. 4 (2002) 115 | DOI

[13] K Cieliebak, A Oancea, Symplectic homology and the Eilenberg–Steenrod axioms, Algebr. Geom. Topol. 18 (2018) 1953 | DOI

[14] K Cieliebak, E Volkov, First steps in stable Hamiltonian topology, J. Eur. Math. Soc. 17 (2015) 321 | DOI

[15] D A Cox, J B Little, H K Schenck, Toric varieties, 124, Amer. Math. Soc. (2011) | DOI

[16] M Entov, L Polterovich, Rigid subsets of symplectic manifolds, Compos. Math. 145 (2009) 773 | DOI

[17] A Fauck, Rabinowitz–Floer homology on Brieskorn manifolds, PhD thesis, Humboldt University of Berlin (2016)

[18] H Geiges, K Zehmisch, The Weinstein conjecture for connected sums, Int. Math. Res. Not. 2016 (2016) 325 | DOI

[19] P Ghiggini, K Niederkrüger, C Wendl, Subcritical contact surgeries and the topology of symplectic fillings, J. Éc. Polytech. Math. 3 (2016) 163 | DOI

[20] Y Groman, Floer theory and reduced cohomology on open manifolds, preprint (2015)

[21] Y Kawamata, D–equivalence and K–equivalence, J. Differential Geom. 61 (2002) 147 | DOI

[22] M Kontsevich, Y Manin, Gromov–Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys. 164 (1994) 525 | DOI

[23] J Latschev, C Wendl, Algebraic torsion in contact manifolds, Geom. Funct. Anal. 21 (2011) 1144 | DOI

[24] Y P Lee, Quantum Lefschetz hyperplane theorem, Invent. Math. 145 (2001) 121 | DOI

[25] E Lerman, Contact toric manifolds, J. Symplectic Geom. 1 (2003) 785

[26] N C Leung, Y Li, Twin Lagrangian fibrations in mirror symmetry, J. Symplectic Geom. 17 (2019) 1331 | DOI

[27] S Luo, Cohomology rings of good contact toric manifolds, preprint (2010)

[28] P Massot, K Niederkrüger, C Wendl, Weak and strong fillability of higher dimensional contact manifolds, Invent. Math. 192 (2013) 287 | DOI

[29] D Mcduff, D Salamon, J–holomorphic curves and symplectic topology, 52, Amer. Math. Soc. (2004) | DOI

[30] D Mcduff, S Tolman, Topological properties of Hamiltonian circle actions, Int. Math. Res. Pap. 2006 (2006) | DOI

[31] M Mclean, Lefschetz fibrations and symplectic homology, Geom. Topol. 13 (2009) 1877 | DOI

[32] A Oancea, Fibered symplectic cohomology and the Leray–Serre spectral sequence, J. Symplectic Geom. 6 (2008) 267 | DOI

[33] D O Orlov, Triangulated categories of singularities and D-branes in Landau–Ginzburg models, Tr. Mat. Inst. Steklova 246 (2004) 240

[34] A F Ritter, Deformations of symplectic cohomology and exact Lagrangians in ALE spaces, Geom. Funct. Anal. 20 (2010) 779 | DOI

[35] A F Ritter, Topological quantum field theory structure on symplectic cohomology, J. Topol. 6 (2013) 391 | DOI

[36] A F Ritter, Floer theory for negative line bundles via Gromov–Witten invariants, Adv. Math. 262 (2014) 1035 | DOI

[37] A F Ritter, Circle actions, quantum cohomology, and the Fukaya category of Fano toric varieties, Geom. Topol. 20 (2016) 1941 | DOI

[38] A F Ritter, I Smith, The monotone wrapped Fukaya category and the open-closed string map, Selecta Math. 23 (2017) 533 | DOI

[39] P Seidel, π1 of symplectic automorphism groups and invertibles in quantum homology rings, Geom. Funct. Anal. 7 (1997) 1046 | DOI

[40] P Seidel, A biased view of symplectic cohomology, from: "Current developments in mathematics, 2006" (editors B Mazur, T Mrowka, W Schmid, R Stanley, S T Yau), International (2008) 211

[41] P Seidel, Fukaya categories and Picard–Lefschetz theory, Eur. Math. Soc. (2008) | DOI

[42] P Seidel, Lagrangian homology spheres in (Am) Milnor fibres via C∗–equivariant A∞–modules, Geom. Topol. 16 (2012) 2343 | DOI

[43] P Seidel, Disjoinable Lagrangian spheres and dilations, Invent. Math. 197 (2014) 299 | DOI

[44] P Seidel, I Smith, A link invariant from the symplectic geometry of nilpotent slices, Duke Math. J. 134 (2006) 453 | DOI

[45] P Seidel, J P Solomon, Symplectic cohomology and q–intersection numbers, Geom. Funct. Anal. 22 (2012) 443 | DOI

[46] N Sheridan, On the Fukaya category of a Fano hypersurface in projective space, Publ. Math. Inst. Hautes Études Sci. 124 (2016) 165 | DOI

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

[48] I Smith, R P Thomas, S T Yau, Symplectic conifold transitions, J. Differential Geom. 62 (2002) 209 | DOI

[49] C Viterbo, Functors and computations in Floer homology with applications, I, Geom. Funct. Anal. 9 (1999) 985 | DOI

[50] A Weinstein, Contact surgery and symplectic handlebodies, Hokkaido Math. J. 20 (1991) 241 | DOI