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On the topology of real Lagrangians in toric symplectic manifolds

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Abstract

We explore the topology of real Lagrangian submanifolds in a toric symplectic manifold which come from involutive symmetries on its moment polytope. We establish a real analog of the Delzant construction for those real Lagrangians, which says that their diffeomorphism type is determined by combinatorial data. As an application, we realize all possible diffeomorphism types of connected real Lagrangians in toric symplectic del Pezzo surfaces.

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Acknowledgements

The authors cordially thank Felix Schlenk for carefully reading the first draft. We also thank the anonymous referee for many useful comments. The paper was carried out when the authors visited the Institut de Mathématiques at Neuchâtel and the Korea Institute for Advanced Study at Seoul. We are grateful for their warm hospitality. JM specially thanks her advisor Suyoung Choi for continued support and encouragement. JB is supported by the grant 200021-181980/1 of the Swiss National Foundation. JK and JM are supported by Samsung Science and Technology Foundation under Project Number SSTF-BA1901-01. JM is partially supported by the National Research Foundation of Korea Grant No. NRF-2021R1A6A3A01087191 and NRF-2019R1A2C2010989.

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Authors and Affiliations

  1. Institut de Mathématiques, Université de Neuchâtel, Rue Emile-Argand 11, 2000, Neuchâtel, Switzerland

    Joé Brendel

  2. Department of Mathematics, Sogang University, 35 Baekbeom-ro, Mapo-gu, Seoul, 04107, Republic of Korea

    Joontae Kim

  3. Research Institute of Mathematics, Seoul National University, 1 GwanAkRo, Gwanak-Gu, Seoul, 08826, Republic of Korea

    Jiyeon Moon

Authors
  1. Joé Brendel
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  2. Joontae Kim
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  3. Jiyeon Moon
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Correspondence to Joontae Kim.

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Brendel, J., Kim, J. & Moon, J. On the topology of real Lagrangians in toric symplectic manifolds. Isr. J. Math. 253, 113–156 (2023). https://doi.org/10.1007/s11856-022-2358-7

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  • DOI: https://doi.org/10.1007/s11856-022-2358-7

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