Symplectic capacities, unperturbed curves and convex toric domains

McDuff, Dusa¹ ; Siegel, Kyler²

¹ Mathematics Department, Columbia University, New York, NY, United States
² Department of Mathematics, University of Southern California, Los Angeles, CA, United States

Geometry & topology, Tome 28 (2024) no. 3, pp. 1213-1285.

Résumé

We use explicit pseudoholomorphic curve techniques (without virtual perturbations) to define a sequence of symplectic capacities analogous to those defined recently by the second author using symplectic field theory. We then compute these capacities for all four-dimensional convex toric domains. This gives various new obstructions to stabilized symplectic embedding problems, which are sometimes sharp.

Détail

DOI : 10.2140/gt.2024.28.1213

Keywords: symplectic capacities, symplectic embeddings, stabilized symplectic embedding problem, toric domains, obstruction bundle gluing, lattice point counts

Bibliographie

[1] F Bourgeois, A Morse–Bott approach to contact homology, from: "Symplectic and contact topology: interactions and perspectives" (editors Y Eliashberg, B Khesin, F Lalonde), Fields Inst. Commun. 35, Amer. Math. Soc. (2003) 55 | DOI

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

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

[4] F Bourgeois, A Oancea, Symplectic homology, autonomous Hamiltonians, and Morse–Bott moduli spaces, Duke Math. J. 146 (2009) 71 | DOI

[5] J Chaidez, B Wormleighton, Lattice formulas for rational SFT capacities, preprint (2021)

[6] K Choi, Combinatorial embedded contact homology for toric contact manifolds, preprint (2016)

[7] K Cieliebak, K Mohnke, Symplectic hypersurfaces and transversality in Gromov–Witten theory, J. Symplectic Geom. 5 (2007) 281 | DOI

[8] K Cieliebak, K Mohnke, Punctured holomorphic curves and Lagrangian embeddings, Invent. Math. 212 (2018) 213 | DOI

[9] D Cristofaro-Gardiner, R Hind, Symplectic embeddings of products, Comment. Math. Helv. 93 (2018) 1 | DOI

[10] D Cristofaro-Gardiner, R Hind, D Mcduff, The ghost stairs stabilize to sharp symplectic embedding obstructions, J. Topol. 11 (2018) 309 | DOI

[11] D Cristofaro-Gardiner, R Hind, K Siegel, Higher symplectic capacities and the stabilized embedding problem for integral elllipsoids [sic], J. Fixed Point Theory Appl. 24 (2022) 49 | DOI

[12] I Ekeland, H Hofer, Symplectic topology and Hamiltonian dynamics, Math. Z. 200 (1989) 355 | DOI

[13] I Ekeland, H Hofer, Symplectic topology and Hamiltonian dynamics, II, Math. Z. 203 (1990) 553 | DOI

[14] M Gromov, Pseudo holomorphic curves in symplectic manifolds, Invent. Math. 82 (1985) 307 | DOI

[15] M Gromov, Soft and hard symplectic geometry, from: "Proceedings of the International Congress of Mathematicians, I" (editor A M Gleason), Amer. Math. Soc. (1987) 81

[16] J Gutt, M Hutchings, Symplectic capacities from positive S1–equivariant symplectic homology, Algebr. Geom. Topol. 18 (2018) 3537 | DOI

[17] R Hind, E Kerman, New obstructions to symplectic embeddings, Invent. Math. 196 (2014) 383 | DOI

[18] R K Hind, E Kerman, J–holomorphic cylinders between ellipsoids in dimension four, J. Symplectic Geom. 18 (2020) 1221 | DOI

[19] M Hutchings, The embedded contact homology index revisited, from: "New perspectives and challenges in symplectic field theory" (editors M Abreu, F Lalonde, L Polterovich), CRM Proc. Lecture Notes 49, Amer. Math. Soc. (2009) 263 | DOI

[20] M Hutchings, Quantitative embedded contact homology, J. Differential Geom. 88 (2011) 231

[21] M Hutchings, Lecture notes on embedded contact homology, from: "Contact and symplectic topology" (editors F Bourgeois, V Colin, A Stipsicz), Bolyai Soc. Math. Stud. 26, Bolyai Math. Soc. (2014) 389 | DOI

[22] M Hutchings, Beyond ECH capacities, Geom. Topol. 20 (2016) 1085 | DOI

[23] M Hutchings, An elementary alternative to ECH capacities, Proc. Natl. Acad. Sci. USA 119 (2022) | DOI

[24] M Hutchings, C H Taubes, Gluing pseudoholomorphic curves along branched covered cylinders, I, J. Symplectic Geom. 5 (2007) 43 | DOI

[25] M Hutchings, C H Taubes, Gluing pseudoholomorphic curves along branched covered cylinders, II, J. Symplectic Geom. 7 (2009) 29 | DOI

[26] D Irvine, The stabilized symplectic embedding problem for polydiscs, preprint (2019)

[27] E Kerman, Y Liang, On symplectic capacities and their blind spots, J. Topol. Anal. (2023) | DOI

[28] M Landry, M Mcmillan, E Tsukerman, On symplectic capacities of toric domains, Involve 8 (2015) 665 | DOI

[29] D Mcduff, A remark on the stabilized symplectic embedding problem for ellipsoids, Eur. J. Math. 4 (2018) 356 | DOI

[30] D Mcduff, K Siegel, Counting curves with local tangency constraints, J. Topol. 14 (2021) 1176 | DOI

[31] A Moreno, R Siefring, Holomorphic curves in the presence of holomorphic hypersurface foliations, preprint (2019)

[32] J Pardon, Contact homology and virtual fundamental cycles, J. Amer. Math. Soc. 32 (2019) 825 | DOI

[33] M B Pereira, Equivariant symplectic homology, linearized contact homology and the Lagrangian capacity, PhD thesis, Universität Augsburg (2022)

[34] M Schwarz, Morse homology, 111, Birkhäuser (1993) | DOI

[35] R Siefring, Intersection theory of punctured pseudoholomorphic curves, Geom. Topol. 15 (2011) 2351 | DOI

[36] K Siegel, Computing higher symplectic capacities, II, in preparation

[37] K Siegel, Higher symplectic capacities, preprint (2019)

[38] K Siegel, Computing higher symplectic capacities, I, Int. Math. Res. Not. 2022 (2022) 12402 | DOI

[39] C Wendl, Automatic transversality and orbifolds of punctured holomorphic curves in dimension four, Comment. Math. Helv. 85 (2010) 347 | DOI

[40] C Wendl, Lectures on symplectic field theory, lecture notes (2020)