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ON TORIC HAMILTONIAN T-SPACES WITH ANTI-SYMPLECTIC INVOLUTIONS

  • Kim, Jin Hong (Department of Mathematics Education Chosun University)
  • Received : 2021.05.24
  • Accepted : 2021.12.31
  • Published : 2022.05.31
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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, ω, µ) be a toric Hamiltonian T-space, and let ∆ = µ(X) denote the moment polytope. Let τ be an anti-symplectic involution of X such that τ maps the fibers of µ to (possibly different) fibers of µ, and let p0 be a point in the interior of ∆. If the toric fiber µ-1(p0) is real Lagrangian with respect to τ, then we show that p0 should be the origin and, furthermore, ∆ should be centrally symmetric.

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Acknowledgement

The author is very grateful to the anonymous referee for providing valuable comments on an earlier version of this paper. This study was supported by research fund from Chosun University (2021).

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