I've read this proposition:
"there is no automorphism (biholomorphic map) of a Kahlerian complex manifold $X$ which acts on $H^2(X,\mathbb{Z})$ as $-$identity"
so I tried to prove this statement and i would like to ask you if I'm correct. Here is as I proceed:
Suppose there is such an automorphism, call it $f$. Then $f^*$ brings a Kahler class $\omega$ to $-\omega$. Writing $\omega$ as $g(I-,-)$, where $g$ is a kahlerian metric, this means that $f^*$ brings $g(I-,-)$ to $-g(I-,-)$. That means that on the second copy of $X$ there should be a riemannian metric $-g$, but it is not an inner product on the tangent bundle (for the positive condition).