a Kähler manifold is symplectic

Let ω(X,Y)=g(JX,Y) on a Kähler manifold. We will prove that ω is a symplectic formMathworldPlanetmath.

  • ω(X,Y)=g(JX,Y)=g(Y,JX)=g(JY,J2X)=g(JY,-X)=-g(JY,X)=-ω(Y,X). Here we used the fact that g is an Hermitian tensor on a Kähler manifold (g(X,Y)=g(JX,JY))

  • ω is linear

    Due to anti-symmetry, we just need to check linearity on the second slot. Since g(JX,) is by definition linear, ω will also be linear.

  • ω is non degenerate

    On a given point on the manifold, pick a non null vector X, αX()=ω(X,)=g(JX,). Since g is non-degenerate11no vector but the null vector is orthogonalMathworldPlanetmathPlanetmathPlanetmath to every other vector, α is also non-degenerate (for all X). ω is thus non degenerate.

  • ω is closed

    First note that

    X(ω(Y,Z)) = X(ω(Y,Z))
    = X(g(JY,Z))
    = g(X(JY),Z)+g(JY,XZ)
    = g(JXY,Z)+g(JY,XZ)
    = ω(XY,Z)+ω(Y,XZ)

    Here we used the fact that both g and J are covariantly constant (g=0 and J=0)

    We aim to prove that dω=0 which is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to proving (dω)(X,Y,Z)=0 for all vector fields X,Y,Z.

    Since this is a tensorial identityPlanetmathPlanetmath, WLOG we can assume that at a specific point p in the Kähler manifold [X,Y]p=[Y,Z]p=[Z,X]p=0 and prove the indentity for these vector fields22in particular this works for the canonical base of TpM associated with a local coordinate system.

    Consider X,Y,Z with the previous commutation relationsMathworldPlanetmathPlanetmath at p, using the formulas for differential forms of small valence:

    (dω)(X,Y,Z) = X(ω(Y,Z))+Y(ω(Z,X)+Z(ω(X,Y)))
    = ω(XY,Z)+ω(Y,XZ)+
    = ω(XY-YX,Z)+ω(YZ-ZY,X)+ω(ZX-XZ,Y)

    The Levi-Civita connectionMathworldPlanetmath is torsion-free, XY-YX=[X,Y] thus:


    And since all the commutators are null at p (by assumptionPlanetmathPlanetmath) we get that:


    ω is therefore closed.

Title a Kähler manifold is symplectic
Canonical name AKahlerManifoldIsSymplectic
Date of creation 2013-03-22 16:07:54
Last modified on 2013-03-22 16:07:54
Owner cvalente (11260)
Last modified by cvalente (11260)
Numerical id 15
Author cvalente (11260)
Entry type Result
Classification msc 53D99
Related topic KahlerManifold