# Grauert's criteria for ample line bundles

In their book "Compact complex surfaces",W,P.Barth,K.Hulek,C.A.M.Peters and Van De Ven refer to the following theorem:

Let $X$ be a compact complex space and $L$ a holomorphic line bundle on $X$.Then $L$ is ample if and only if the folowing holds:given any irreducible analytic subset $Y$ of strictly positive dimension on $X$,there exist an $n=n(Y)$,such that $L^{\otimes n} |_Y$ has a section which has at least one zero,but dose not vanish identically.

They don't give a proof in their book.Instead they refer to Grauert's original paper wich was published in Math.Ann. 1962.(I'm sorry I do not know how to type the title of the paper because it is written in German)

I do not know German.So my question is:Can I find the proof of this theorem somewhere else? Or instead,some comments on the idea of proof will also be very helpfull.

At last,this paper of Grauert is among one of the papers I want to read with greatest euthusiasm.Dose there exist a translation?Or Can I find some books or papers which gives an explanation of the results of this paper?

Thank you in advance!

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## 1 Answer

You can find a proof in Kleiman's famous paper Toward a Numerical Theory of Ampleness, Theorem 1 page 317.

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Kleiman states that his proof is for complete algebraic schemes. Does it work verbatim for compact analytic spaces? – Ben McKay Sep 21 '11 at 8:55
I had a (quick) look at the proof and it seems to me that it should extend without problems to the compact analytic setting. – Francesco Polizzi Sep 21 '11 at 9:16
The reference is very helpful.Thank you. – Jun Li Sep 21 '11 at 12:02
even if you think you don't read german, you might try Grauert's paper, in pretty clear german, from middle page 347 to top 349. he uses induction, finds X1, the zero set in X of a section of a power of F, then applies induction to the restriction of F to X1. If Kleiman uses the same argument you might read them together. Grauert uses Remmert's reduction theorem which allows one to blow down compact subvarieties of a holom convex space to a Stein space, with the structure sheaf also pushing down. gdz.sub.uni-goettingen.de/en/dms/load/img – roy smith Sep 22 '11 at 20:26
verschwinden = vanishes, schnittfla"che = section, keine = no, Menge = set, Teilmenge = subset, Garbe = sheaf, Geradenbundel = line bundle, Jede = any, fortsetzen =(?) extends, and you can guess: exakte Sequenz = .... – roy smith Sep 22 '11 at 20:35