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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.

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    $\begingroup$ An alternative argument is to use the asymptotics of (complex analytic) Riemann-Roch for the integer-valued function $\chi(S,\mathcal{O}_S(nD))$ of $n$, where $D$ is an effective divisor whose support is contained in $C$. The leading term is the self-intersection of $D$ on $S$. However, it is straightforward to see that both $h^0$ and $h^2$ do not grow more than linearly in $n$. Thus, the self-intersection must be nonpositive. $\endgroup$
    – Jason Starr
    Commented Jul 10 at 8:59

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  1. All the components of $C$ are orthogonal to the pullback $f^*H_T$ of an ample class of $T$. One has $$ (f^*H_T, f^*H_T) = (H_T,H_T) > 0, $$ hence its orthogonal complement is negative definite.

  2. A generalization to higher dimension is given by Mori contraction theorem, see https://encyclopediaofmath.org/wiki/Mori_theory_of_extremal_rays.

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    $\begingroup$ The OP is working in the category of compact complex analytic spaces, some of which are not even Kaehler, much less Moishezon. There may not be any ample classes (although, since the problem is analytic local on the target, we should be able to “transport” to the projective setting). $\endgroup$
    – Jason Starr
    Commented Jul 9 at 8:26
  • $\begingroup$ @JasonStarr Could you explain how to transport it to projective setting? $\endgroup$
    – Hydrogen
    Commented Jul 9 at 17:28
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    $\begingroup$ @Hydrogen: As soon as Hodge index theorem holds and there is a class with positive square on $T$, the same argument works. $\endgroup$
    – Sasha
    Commented Jul 9 at 17:48
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    $\begingroup$ @Hydrogen “Could you explain how to transport it to the projective setting?” The morphism $f$ is a (weakly) projective morphism, so locally (analytically) it is a blowing up of an ideal sheaf on $T$. Although it is a sledgehammer, we can use Artin’s approximation theorem to find an algebraic surface singularity germ and an ideal sheaf on that germ that agrees to arbitrary order with the original ideal sheaf on $T$. $\endgroup$
    – Jason Starr
    Commented Jul 9 at 18:57
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    $\begingroup$ The morphism $f$ is only "weakly" projective, i.e., projective analytically locally on $T$ (good enough for our purpose). For each irreducible component $C_i$ of $C$, you can find a germ of an analytic arc $D_i$ that intersects $C_i$ transversally at one point not on any curve $C_j$. The arc $D_i$ defines an (analytic) invertible sheaf on an analytic neighborhood of $C$ in $S$. When you tensor all of these analytic invertible sheaves together (for all components $C_i$ of $C$), you get an $f$-ample analytic invertible sheaf on an analytic neighborhood of $C$. $\endgroup$
    – Jason Starr
    Commented Jul 9 at 19:21

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