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!