One direction of the Grauert's contractibility theorem shows
Let $f:S\rightarrow T$ be a surjective holomophic map where $S$ is a compact holomorphic surface. If $C$ is a reduced connected effective divisor such that $f(C)$ is a point, then the intersection matrix of irreducible components of $C$ is negative definite.
I have some questions:
It is claimed that this is a special case of the Hodge index theorem. I'm wondering how to see it? Hodge index theorem shows the index is $(1,h^{1,1}(S)-1)$, but I have no idea how to deal with the first positive index.
Is it possible to generalize it to higher dimensional complex manifolds? I know another direction of Grauert's theorem is hard, but if only for the negative definite part, is it generalizable?
