# Bulletin of theKorean Mathematical SocietyBKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc.

Published online March 8, 2022

## On toric Hamiltonian T-spaces with anti-symplectic involutions

Jinhong Kim

Chosun University

### Abstract

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 $X$ 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. In this paper, we also provide a simple example asserting that the condition of $\tau$ preserving the fibration structure of $\mu$ plays a crucial role in the proof of our main result, which thus disproves a general question stated without any restriction about the fibration structure of $\mu$.

Keywords: Hamiltonian $T$-spaces, anti-symplectic involutions, moment polytopes, conjugations, quasitoric manifolds, small covers, real Lagrangians

MSC numbers: 35D12