Abstract
In this paper we establish a topological constraint for monotone Lagrangian embeddings in certain complex hypersurfaces of integral Kähler manifolds. As an application, we prove that it is impossible to embed a connected sum of \(S^{1}\times S^{2k}\)s in \(\mathbb {C}P^{2k+1}\) as a monotone Lagrangian.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arnold, V.I.: Une classe caractéristique intervenant dans les conditions de quantification. Funktsional Anal. i Prilozhen. 1, 1–14 (1967)
Biran, P.: Lagrangian barriers and symplectic embeddings. Geom. Funct. Anal. GAFA 11(3), 407–464 (2001). doi:10.1007/PL00001678
Biran, P.: Lagrangian non-intersections. Geom. Funct. Anal. GAFA 16(2), 279–326 (2006). doi:10.1007/s00039-006-0560-0
Biran, P., Cieliebak, K.: Lagrangian embeddings into subcritical Stein manifolds. Isr. J. Math. 127(1), 221–244 (2002). doi:10.1007/BF02784532
Biran, P., Khanevsky, M.: A Floer–Gysin exact sequence for Lagrangian submanifolds. Comment. Math. Helvetici 88(4), 899–952 (2013)
Bourgeois, F., Eliashberg, Y., Hofer, H., Wysocki, K., Zehnder, E.: Compactness results in symplectic field theory. Geom. Topol. 7(2), 799–888 (2003)
Cieliebak, K., Eliashberg, Y.: From Stein to Weinstein and Back: Symplectic Geometry of Affine Complex Manifolds. American Mathematical Society colloquium publications. American Mathematical Society (2012). https://books.google.fr/books?id=In1Dbj-pkIkC
Damian, M.: Floer homology on the universal cover, a proof of Audin’s conjecture and other constraints on Lagrangian submanifolds. Comment. Math. Helvetici 87(2), 433–462 (2012)
Damian, M.: On the topology of monotone Lagrangian submanifolds (2012). http://arxiv.org/abs/1206.2676v1. To be published in the Annales scientifiques de l’École normale supérieure
Eliashberg, Y., Givental, A., Hofer, H.: Introduction to symplectic field theory. GAFA - Special Volume, Part II, 560–673 (1999)
Fukaya, K.: Application of Floer homology of lagrangian submanifolds to symplectic topology. “Morse Theoretic methods in Nonlinear Analysis and in Symplectic Topology”, P. Biran (ed.) etc. Nato Science Series II 217, 231–276 (2005)
Gromov, M.: Pseudo holomorphic curves in symplectic manifolds. Invent. Math. 82(2), 307–347 (1985). doi:10.1007/BF01388806
McDuff, D., Salamon, D.: J-holomorphic curves and symplectic topology. No. vol. 52 in American mathematical society colloquium publications. American Mathematical Society (2004). http://books.google.fr/books?id=jCd0-pzGo5AC
Nemirovski, S.: Lagrangian Klein bottles in \(\mathbb{R}^{2n}\). Geom. Funct. Anal. GAFA 19(3), 902–909 (2009). doi:10.1007/s00039-009-0014-6
Acknowledgments
I want to thank Mihai Damian for his incredible patience as he guided me through the writing of this article. I also wish to thank Michael Khanevsky and Romain Ponchon for the time they took to discuss various parts of it. Thanks to Florian Delage for his kind proofing. And finally, I would like to thank the anonymous referee for their abundant and relevant remarks.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Schatz, S. A topological constraint for monotone Lagrangians in hypersurfaces of Kähler manifolds. Math. Z. 281, 877–892 (2015). https://doi.org/10.1007/s00209-015-1511-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-015-1511-y