TITLE: What does the Lefschetz principle (in algebraic geometry) mean exactly?
QUESTION [36 upvotes]: 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:)

REPLY [27 votes]: 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.