• Title/Summary/Keyword: anti-symplectic involution

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ANTI-SYMPLECTIC INVOLUTIONS ON NON-KÄHLER SYMPLECTIC 4-MANIFOLDS

  • Cho, Yong-Seung;Hong, Yoon-Hi
    • Journal of the Korean Mathematical Society
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    • v.44 no.4
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    • pp.757-766
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    • 2007
  • In this note we construct an anti-symplectic involution on the non-$K\ddot{a}hler$, symplectic 4-manifold which is constructed by Thurston and show that the quotient of the Thurston's 4-manifold is not symplectic. Also we construct a non-$K\ddot{a}hler$, symplectic 4-manifold using the Gomph's symplectic sum method and an anti-symplectic involution on the non-$K\ddot{a}hler$, symplectic 4-manifold.

ON TORIC HAMILTONIAN T-SPACES WITH ANTI-SYMPLECTIC INVOLUTIONS

  • Kim, Jin Hong
    • Bulletin of the Korean Mathematical Society
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    • v.59 no.3
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    • pp.671-683
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    • 2022
  • 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.

A NOTE ON INDECOMPOSABLE 4-MANIFOLDS

  • Cho, Yong-Seung;Hong, Yoon-Hi
    • Bulletin of the Korean Mathematical Society
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    • v.42 no.4
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    • pp.817-828
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    • 2005
  • In this note we show that there is an anti-symplectic involution $\sigma\;:\;X\;\to\;X$ on a simply-connected, closed, non-Kahler and symplectic 4-manifold X with a disjoint union of Riemann surfaces ${\amalg}^n_{i=1}{\Sigma}_i,\;n\;{\ge}\;2$ as a fixed point set. Also we show that its quotient X/$\sigma$ is homeomorphic to $\mathbb{CP}^2{\sharp}r\mathbb{CP}^2$ but not diffeomorphic to $\mathbb{CP}^2{\sharp}r\mathbb{CP}^2,\;r\;=\;b_2^-(X/{\sigma})$.

COUNTING REAL J-HOLOMORPHIC DISCS AND SPHERES IN DIMENSION FOUR AND SIX

  • Cho, Cheol-Hyun
    • Journal of the Korean Mathematical Society
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    • v.45 no.5
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    • pp.1427-1442
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    • 2008
  • We provide another proof that the signed count of the real J-holomorphic spheres (or J- holomorphic discs) passing through a generic real configuration of k points is independent of the choice of the real configuration and the choice of J, if the dimension of the Lagrangian submanifold L (fixed point set of involution) is two or three, and also if we assume L is orient able and relatively spin. We also assume that M is strongly semi-positive. This theorem was first proved by Welschinger in a more general setting, and we provide more natural approach using the signed degree of an evaluation map.