This principle claims that every true statement about a variety over the complex number field $\mathbb{C}$ is true for a variety over any algebraic closed field of characteristic 0.


But what is it mean? Is there some "statement" not allowed in this principle?


Is there an analog in char p>0?


Is there reference about this topic? I tried to find some but in vain.


Thanks:)



What does the Lefschetz principle (in algebraic geometry) mean exactly?
The Lefschetz principle was formulated and illustrated the first time in:



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> S. Lefschetz, *Algebraic Geometry*, Princeton University Press, 1953.
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The basic idea is that every equation over some algebraically closed field of characteristic $0$ only involves finitely many elements, which generate a subfield isomorphic to a subfield of $\mathbb{C}$. But as Seidenberg points out in



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> A. Seidenberg, *Comments on Lefschetz's principle*, American American Monthly (65), No. 9, Nov. 1958, 685 - 690
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Lefschetz has not given a rigorous proof and it is not clear at all if it holds when analytical methods over $\mathbb{C}$ are used. Tarski's classical result that the theory of algebraically closed fields of characteristic $0$ admits quantifier elimination and therefore all models are elementary equivalent is called the "Minor Lefschetz principle", because it does not apply to prominent examples such as Hilbert's Nullstellensatz.


A precise formulation, with a short proof, which works in *every* characteristic, can be found here:



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> Paul C. Eklof, *Lefschetz's Principle and Local Functors*, Proc. AMS (37), Nr. 2, Feb. 1973, [online](http://www.ams.org/journals/proc/1973-037-02/S0002-9939-1973-0325389-7/S0002-9939-1973-0325389-7.pdf)
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In the language of that paper, the principle states the following: Let $F$ be a functor from universal domains of characteristic $p$ ( = algebraically closed field of infinite transcendence degree over $\mathbb{F}\_p$) to some category of many-sorted structures with embeddings, which satisfies the following finiteness condition: If $K \subseteq L$ is an extension, then every finite subset of $F(L)$ is already included in the image of a subextension of finite transcendence degree over $K$.


Then, for all $K,L$, we have that $F(K)$ and $F(L)$ are elementary equivalent.


For a specific statement one wants to prove using the Lefschetz princple, one can take $F(K)$ to be the collection of all "relevant algebraic geometry over $K$".


A generalization is treated in:



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> Gerhard Frey, Hans-Georg Rück, *The strong Lefschetz Principle in Algebraic Geometry*, manuscripta math. (55), 385 - 401 (1986)
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