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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:

  1. 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.

  2. 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?

Edit: Assume $S,T$ to be Kahler if it makes things better.

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:

  1. 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.

  2. 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?

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:

  1. 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.

  2. 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?

Edit: Assume $S,T$ to be Kahler if it makes things better.

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Negative Definite of Exceptional Curve in Higher Dimension

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:

  1. 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.

  2. 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?