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Exact Lagrangian immersions with a single double point


Authors: Tobias Ekholm and Ivan Smith
Journal: J. Amer. Math. Soc. 29 (2016), 1-59
MSC (2010): Primary 53D35, 53D40, 14J70; Secondary 14N05
DOI: https://doi.org/10.1090/S0894-0347-2015-00825-6
Published electronically: January 9, 2015
MathSciNet review: 3402693
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Abstract: We show that if a closed orientable $ 2k$-manifold $ K$, $ k>2$, with Euler characteristic $ \chi (K)\ne -2$ admits an exact Lagrangian immersion into $ \mathbb{C}^{2k}$ with one transverse double point and no other self-intersections, then $ K$ is diffeomorphic to the sphere. The proof combines Floer homological arguments with a detailed study of moduli spaces of holomorphic disks with boundary in a monotone Lagrangian submanifold obtained by Lagrange surgery on $ K$.


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Tobias Ekholm
Affiliation: Department of Mathematics, Uppsala University, Box 480, Uppsala 751 06, Sweden; and Institut Mittag-Leffler, Aurav. 17, Djursholm 182 60, Sweden
Email: tobias.ekholm@math.uu.se

Ivan Smith
Affiliation: Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, United Kingdom
Email: is200@cam.ac.uk

DOI: https://doi.org/10.1090/S0894-0347-2015-00825-6
Received by editor(s): November 25, 2011
Received by editor(s) in revised form: July 11, 2014, and September 6, 2014
Published electronically: January 9, 2015
Additional Notes: The first author was partially supported by the Knut and Alice Wallenberg Foundation, as a Wallenberg Scholar.
The second author was partially supported by European Research Council grant ERC-2007-StG-205349.
Article copyright: © Copyright 2015 American Mathematical Society