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

S. Lefschetz, Algebraic Geometry, Princeton University Press, 1953.

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

A. Seidenberg, Comments on Lefschetz's principle, American American Monthly (65), No. 9, Nov. 1958, 685 - 690

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:

Paul C. Eklof, Lefschetz's Principle and Local Functors, Proc. AMS (37), Nr. 2, Feb. 1973, online

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:

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

A: I'm not sure I should admit this in public, but although I am aware of the precise formulations using first order logic and beyond (mentioned in the above answers), 
I tend not to use them. Rather I view the Lefschetz principle as more of a philosphical principle of what ought to be possible in general, and do the necessary verifications as and when I need them (but perhaps  only implicitly). 
I suspect this attitude is pretty common among many algebraic geometers.
To give an example,
for many years the only known proofs* of the Kodaira vanishing theorem were analytic.
But since coherent cohomology behaves well under field extensions, Kodaira vanishing 
is valid over arbitrary (not necessarily algebraically closed) fields of characteristic $0$.
On the other hand, for certain kinds of arguments, one needs a big enough field to
carry out the argument. This typically happens when one is forced to remove a countable
union of exceptional sets.  Curiously, the Noether-Lefschetz theorem is one
such case. Here the Lefschetz principle in the most naive sense
won't work.
*(Added Footnote.) There is now an algebraic proof due to Deligne and Illusie, which involves
reduction to positive characteristic. This is yet another kind of transfer.
A: For extensions going well beyond first-order logic, and allegedly covering the uses of the Lefschetz principle that one wants to make in algebraic geometry, see "Lefschetz's principle" by Jon Barwise and Paul Eklof (J. Algebra 13 (1969) 554-570; Math Reviews 41 #5207).
