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Let $K$ be an algebraically closed field of characteristic $0$. Let $P$ be a proposition on a non-singular projective variety over $K$ which is stated in the language of algebraic geometry. Suppose $P$ holds when $K = \mathbb{C}$. Does $P$ hold on any such $K$?

Remark We may take as $P$, for example, the Kodaira vanishing theorem.

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    Don't know what "language of algebraic geometry" means. If we restrict to the language of fields, then since any two algebraically closed fields of characteristic $0$ are elementarily equivalent, we can mechanically transfer.2012-12-03
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    @AndréNicolas Please read the link of the Kodaira vanishing theorem. It is a result on a compact Kähler manifold and a positive holomorphic line bundle on it. However, by the GAGA principle, it can be translated into the result of a projective non-singular variety and an ample invertible sheaf on it. For a long time(until 1987) it was the only proof of the algebraic geometric version of the theorem. But GAGA makes sense only over $\mathbb{C}$.2012-12-03
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    I know almost nothing of algebraic geometry, but there are quite a few model-theoretic results of this general kind, where one uses special tools to prove a theorem in a certain setting, and then transfer machinery, such as Compactness, or something more subtle.2012-12-03
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    @AndréNicolas My point is that even though a result can be stated algebraically, it is not clear if it can be proved algebraically, because on $\mathbb{C}$ we can use powerful analytic methods.2012-12-03
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    If it is a sentence in the theory of algebraically closed fields of characteristic $0$, then it can be proved "algebraically," since the theory is decidable. At its crudest, start listing all proofs. sooner or later $\phi$ or $\lnot\phi$ will show up in the list. (There are better approaches!) By the way, there is a similar theorem for real-closed fields, so one can also handle, in this limited sense, real algebraic geometry.2012-12-03
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    @AndréNicolas That is interesting. So if a theorem on $\mathbb{C}$ which is stated algebraically like the Kodaira vanishing theorem is proved to be true, it can be proved algebraically, right? Hence it holds on any field of characteristic $0$, right?2012-12-03
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    @Makoto: Our notions of stated algebraically may be very different. I mean using plus, times, logical symbols. Can't even quantify over polynomials, though can for polynomials of explicit degree $\le 77$. And not "any field of characteristic $0$", any *algebraically closed* field of characteristic $0$.2012-12-03
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    I noticed that someone serially upvoted for my questions and answers including this one. While I appreciate them, I would like to point out that serial upvotes are automatically reversed by the system.2013-11-27

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