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On a complex manifold $M$ a Kähler form is a symplectic form $\omega$ which is compatible with the canonical almost complex structure $J$ in the following sense $$\omega({}\cdot{},J{}\cdot{})$$ is a Riemannian metric tensor, i.e. symmetric and positive definite.

From $J^*\omega=\omega$ we have $\omega\in\Omega^{1,1}$ then I find the following weird $$\omega(\partial_{z_1},J \partial_{z_1})= i\omega(\partial_{z_1}, \partial_{z_1}) =0$$ because $\omega\in\Omega^{1,1}$ but it should be positive .

Can someone tell me where I am wrong?

Gibbs
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Display Name
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1 Answers1

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You are not wrong. You can double check this as follows: $$g(\partial_z,\partial_z) = \frac{1}{4}g(\partial_x-i\partial_y,\partial_x-i\partial_y) = \frac{1}{4}(1-1)=0.$$ The metric $g$ I use here is the Riemannian metric on your manifold extended by $\mathbb{C}$-linearity on the complexified tangent bundle. This suggests you that this metric is actually of type $(1,1)$.

Gibbs
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    I guess the problem is that if we only test by vectors but not complexified vectors, then we can get positive definite. – Display Name May 27 '18 at 19:55