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I am interested in reading the proof of Grauert's Contractibility Theorem, asserting that an integral compact curve in a smooth compact surface (without the projectivity assumption - this is the case I am mostly interested) is exceptional iff it has negative self-intersection.

The reference is this: Grauert, H.: Uber Modifikationen und exzeptionelle analytischen Mengen, Math. Ann. 146 (1962), 331-368.

Unfortunately I do not read german; all the sources I tried only state the result without proof (e.g. Barth-Hulek-Peters-Van de Ven) or the prove the algebraic version.

Does anyone know a reference where I can read the proof, or at least the main ideas or main steps?

Thank you.

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1 Answer 1

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Fujiki has proved (in English!) the following generalization of Grauert Theorem in the analytic category. Let $X$ be a complex space, $A$ be an effective Cartier divisor on $X$ and $f:A \longrightarrow A'$ be a proper surjective morphism. Let $L =[A]$ be the line bundle corresponding to $A$. Assume the following conditions:

_$L^*|_A$ is $f$-ample,

_$R^1f_*(L^*|_A)^{\otimes m} = 0$ for every $m>0$.

Then, there exists a birational morphism $g : X \longrightarrow X'$ such that $g|_A = f$. See Theorem 1 page 495 of Fujiki's paper

In case $X$ is a compact surface and $A$ is a compact smooth (=integral) curve, the first hypothesis is equivalent to the self-intersection of the curve being negative by the normal bundle formula. The second hypothesis is always satisfied (provided that $L^*|_A$ is ample) if $A$ is a compact smooth curve by Serre duality and the fact the a negative line bundle on a smooth compact curve has no sections.

The result above by Fujiki is the exact translation of the same result proved by M. Artin in the category of algebraic spaces. It seems however the proof in the analytic category needs some extra-arguments, as Fujiki relies on the theory of plusri-subharmonic functions.

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  • $\begingroup$ Thank you very much for this answer! Fujiki's result looks pretty nice indeed. Anyway, it looks like this is not really a generalisation of Grauert's result, since here one starts with a surjective map from the divisor; while Grauert seems to prove that having negative-self intersection implies the existence of such a surjection (but probably in the higher dimensional case this is needed)... $\endgroup$
    – User3773
    Nov 19, 2019 at 15:21
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    $\begingroup$ @User3773 In the case of curve in a surface, there is always certainly a proper surjective map from a curve to... a point $\endgroup$
    – Libli
    Nov 19, 2019 at 16:39

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