I have the following question
$X$- a compact Kahler manifold and $v\in \Gamma(X,TX)$- Killing vector field. I don't understand why Lie derivatives equal to zero
$L_v \omega=0$ where $\omega$ is the Kahler form
$L_vI=0$ where $I$ is the complex structure
My attempt is the following. Killing field is satisfied the following equation $L_vg=0$. We have known from the definition that A Kahler Manifold is an hermitian manifold, whose Kälher form is closed. It follows that $d\omega=0$. Let $h$-hermitian metric and from the famous theorem we know that there is a correspondence between $h \to \omega=-Im h$. And we have that $g(m,n)=\omega(m,In)$, where $I$ is the complex structure. And we can obtain the result or am I a wrong about the first statement?
I don't know what to do with the second statement. If you don't mind please explain it in more details. Thank you!