# What´s essential to learn about complex spaces and several complex variables for an algebraic geometer?

Hi, I don´t know if this question is suitable for this site. The field of several complex variables is too broad, so I would like to know what´s essential to learn about complex spaces and several complex variables for an algebraic geometer? Any references?

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Griffiths-Harris: "Principles of algebraic geometry", Voisin: "Hodge theory and complex algebraic geometry", Huybrechts: "Complex geometry". The material covered in these books is more than enough in order to get started. –  Francesco Polizzi Jun 10 at 13:45
Just this Francesco? :p –  diverietti Jun 10 at 21:20
I said "to get started" :-) –  Francesco Polizzi Jun 11 at 8:36

There is the famous GAGA of J.P.Serre . But here is an elementary introduction (a master thesis !) : Elementary GAGA , by Kay Werndli, University of Basel (26th July, 2011).

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May I insist ? : Elementary GAGA , by Kay Werndli, University of Basel (Supervisor : H.Kraft), is an excellent easy introduction (21 pages !) to a study and a comparison between GA (géométrie algébrique) and GA (géométrie analytique ). This has nothing to do with the brightly coloured famous Lady. Key words there are : analytification, finite morphism, proper morphism, étale morphism,irreducibility.

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You don't need to repeat yourself. It would be useful to post a link to the cited thesis, if such a thing exists, though. –  Ketil Tveiten Jun 21 at 9:28
Of course the link exists , in PDF , and is easy to find . Go to Google : Elementary GAGA - Mathématiques –  Al-Amrani Jun 21 at 10:43
Thank you for the more elementary suggestion. I started to post the link, but decided against it since one must really recommend Serre's original paper as first option. I suggest one would be better off to at least look at the actual GAGA before taking the other road. –  roy smith Jun 21 at 13:41
There is no contradiction between the two options. It depends on your purpose and on, how much you know already. A link to a good translation of GAGA has been given 3 years ago at the end of : "Serre's FAC in English" - MathOverflow !! –  Al-Amrani Jun 21 at 15:05
The basic yoga of positivity in complex geometry is that ampleness of a line bundle $L$ is equivalent to the positivity of the curvature form of a smooth hermitian metric on $L$. This allows us to treat global algebraic questions involving ampleness and cohomology by looking at pointwise estimates of positive differential forms on our manifold. Once there, all of the machinery of Riemannian and complex geometry is available and hard global questions get converted into extremely computationally messy problems of linear algebra. For certain things, like cohomology of adjoint bundles $K_X \otimes L$, these methods work very well, for others they work less well or not at all, but it's always good to have another tool with which to attack problems.