Bull. Korean Math. Soc. 2022; 59(3): 671-683
Online first article March 8, 2022 Printed May 31, 2022
https://doi.org/10.4134/BKMS.b210406
Copyright © The Korean Mathematical Society.
Jin Hong Kim
Chosun University
The aim of this paper is to deal with the realization problem of a given Lagrangian submanifold of a symplectic manifold as the fixed point set of an anti-symplectic involution. To be more precise, let $(X, \omega, \mu)$ be a toric Hamiltonian $T$-space, and let $\Delta=\mu(X)$ denote the moment polytope. Let $\tau$ be an anti-symplectic involution of $X$ such that $\tau$ maps the fibers of $\mu$ to (possibly different) fibers of $\mu$, and let $p_0$ be a point in the interior of $\Delta$. If the toric fiber $\mu^{-1}(p_0)$ is real Lagrangian with respect to $\tau$, then we show that $p_0$ should be the origin and, furthermore, $\Delta$ should be centrally symmetric.
Keywords: $T$-spaces, anti-symplectic involutions, moment polytopes, conjugations, quasitoric manifolds, small covers, real Lagrangians
MSC numbers: 53D12, 53D20
Supported by: This study was supported by research fund from Chosun University (2021).
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