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The title is too broad. I will narrow it down shortly. My motivation is this. I am interested in complex geometry, and it seems there are (at least) 2 schools having their own notations, methods and so on, to study the same objects: scheme-theoretic (as in R. Hartshorne's "Algebraic Geometry") and complex-analytic (as in Griffiths and Harris's book). I understand that each has its own merits, and this is not really the point of my post.

I am interested in some possible dictionary to translate results from one language to the other. Already there are differences on the level of topological spaces: schemes make use of the Zariski topology, while complex manifold theory uses the manifold topology. There are also differences between the structure sheaves: schemes, in the context of complex varieties, make use of sheaves that are sheaves of germs of regular rational functions, while in the complex analytic school, one uses the sheaf of germs of holomorphic functions as structure sheaf.

Now, I will narrow it down. Consider say Grothendieck's vanishing theorem on one hand, and various vanishing theorems such as the Kodaira-Nakano vanishing theorems and its various generalizations. The conclusions are similar, namely some higher sheaf cohomologies vanish in both cases, but the hypotheses are different: Grothendieck uses Noetherian topological spaces (complex varieties are Noetherian with respect to the Zariski topology, and not the manifold topology), while Kodaira-Nakano use the manifold topology. Maybe my question is naive but:

Question 1: can one deduce one of these vanishing theorems from the other?

Question 2: are there other examples of theorems which have similar conclusions, but have 2 different versions corresponding to the 2 schools?

Question 3: is there a dictionary that allows to translate statements of theorems from one school into the other, so to speak?

Question 1 may be too naive, and perhaps false. Question 2 is perhaps not too specific, but interesting, in my opinion. Question 3 would be ultimately what I would like to "have", assuming such a dictionary exists. If it exists, it may not be so "clean cut" so to speak, because many of the definitions in the scheme-theoretic school were done essentially by algebraists, while those in the complex analytic school were mostly done by geometers and analysts.

Edit: concerning question 1), as Bertram pointed out, the 2 vanishing theorems that I chose were quite different in character, so that was a poor choice. Concerning question 3), Bertram kindly referred me to Serre's GAGA paper, which establishes the dictionary I am seeking, and does it with precise results. I kept hearing about this paper as being a masterpiece. Now I have (finally) decided that it was time to read it!

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    $\begingroup$ The general answer to your question, in particular Q3, is GAGA. (Try searching for "Serre GAGA".) $\endgroup$
    – bertram
    May 30, 2017 at 19:02
  • $\begingroup$ @bertram Thank you! Serre GAGA is one of those papers that are must-read, though I have only heard the results there being paraphrased. I have now finally decided I should definitely read this seminal paper! $\endgroup$
    – Malkoun
    May 30, 2017 at 19:05
  • $\begingroup$ As for Q3: these two theorems are fundamentally different in nature. Grothendieck's theorem says something about cohomology in degrees above the dimension, and applies to any sheaf of abelian groups. Kodaira's theorem is a specialised result that only applies to inverses of ample line bundles, but the conclusion is much stronger: it gives vanishing of cohomology in all degrees (except the top). There is no way to deduce one from the other. $\endgroup$
    – bertram
    May 30, 2017 at 19:11
  • $\begingroup$ @bertram yes, so this was a bad example. I will edit my post. Thank you! $\endgroup$
    – Malkoun
    May 30, 2017 at 19:14

1 Answer 1

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As User bertram pointed out, the main question contained in my post (Question 3), was already asked and answered by Jean-Pierre Serre in his famous GAGA paper.

I will leave this as an answer, in case someone just like me likes to use MSO and online resources in general, and happens to be asking the same question!

Edit: after reading a good part of Serre's GAGA, one can say the following. Due to the manifold topology being finer than the Zariski one, and because a regular rational function is also a holomorphic function, one can define a functor which takes an algebraic variety and maps it into an analytic space. While in general, even for affine varieties, the two categories of algebraic varieties and analytic spaces are certainly non-isormophic, for the special case of projective varieties, the functor is particularly nice: in some sense, one has a very precise dictionary between, on one hand, complex projective varieties, with the Zariski topology, and sheaves of regular (algebraic) functions on them and, on the other hand, their "analytizations", so to speak, which are endowed with the "usual" topology, and the sheaf of holomorphic functions.

Remark: if you haven't read Serre's GAGA, and are sufficiently fluent in French, I strongly advise you to read it.

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    $\begingroup$ Additional useful references for people most familiar with the algebraic point of view include Appendix B in Hartshorne's Algebraic Geometry as a start and Raynaud's SGA 1 Exposé (XII) Géométrie algébrique et géométrie analytique. The latter can serve very well for references of comparison result. $\endgroup$
    – Ben
    May 31, 2017 at 9:45

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