# 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 '13 at 13:45
Just this Francesco? :p –  diverietti Jun 10 '13 at 21:20
I said "to get started" :-) –  Francesco Polizzi Jun 11 '13 at 8:36

If you want a specific goal to understand, one that has connections to most of modern complex geometry and provides motivation for many of the techniques used there, I suggest the link between ampleness in algebraic geometry and positivity in complex geometry. The first applications of these ideas were Kodaira's embedding theorem, which shows that for line bundles positivity is equivalent to ampleness, and the various associated vanishing theorems. Both are excellent starting points and provide managable goals for study.

They are all covered in the books Francesco mentioned, perhaps most thouroughly in Griffiths-Harris. I would also recommend Demailly's Complex analytic and differential geometry, whose Chapters 5-7 go into great detail on these matters (without superfluous discussion, that you may want to get elsewhere, like in Griffiths-Harris, Huybrechts or Voisin's books or Lazarsfeld's "Positivity" series). Once you feel comfortable with this you can look at Demailly's Analytic methods in algebraic geometry, which gives a very modern snapshot of the complex-geometric notions of algebraic geometry and their applications. (Beware, that text is hardcore.)

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.

As I said, historically the first major applications of these methods were the Kodaira-Nakano vanishing theorems and the Kodaira embedding theorem. A more recent, quite impressive, application is Siu's proof of the invariance of plurigenera of algebraic manifolds of general type. His proof makes great use of the modern versions of positivity and currently there is no algebraic proof of the same result. You can find it in his preprints on the arXiv or at the end of Demailly's "Analytic methods" manuscript, but again, this is hard.

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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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