# 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?

Thanks:)

• en.wikipedia.org/wiki/… – Charles Matthews Mar 8 '12 at 10:30
• There are some obvious statements not allowed in this principle--for instance, "the cardinality of every variety is at most that of the continuum" (which is true over $\mathbb{C}$, but not over an algebraically closed field with cardinality greater than that of $\mathbb{C}$). Part of the interest in attempts to formalize the Lefschetz Principle are attempts to formulate exactly what statements should be allowed. – Charles Staats Mar 8 '12 at 14:59
• Another comment is that, while there exist formal versions of the Lefschetz Principle, it is often used as a "proof technique" rather than a theorem to be quoted. For instance, proofs can sometimes proceed in the following manner: 1) Show that the statement you care about can, in any given case, be reduced to a statement over a countable extension of $\mathbb{Q}$, by using the "fact" that the statement only involves countably many elements of your field. 2) Any countable extension of $\mathbb{Q}$ can be embedded in $\mathbb{C}$. – Charles Staats Mar 8 '12 at 15:05
• @Charles: Isn't this a little bit unfortunate because there is a precise theorem which one might quote (the one by Eklof)? In practice one then has to check only 1), whereas 2) (and 3), namely the reason why this does the job) may be omitted. – Martin Brandenburg Mar 8 '12 at 16:03
• [ct'd] Since many algebraic geometers have not taken the time to master the basic techniques for such "translations," applying the precise theorem when the imprecise version can easily be made to work will unnecessarily reduce the accessibility of one's writing. – Charles Staats Mar 11 '12 at 16:34

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)

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.

• The algebraic proof of Kodaira vanishing that you mention is actually due to Raynaud (although the proof is given in the paper of Deligne and Illusie). – ulrich Mar 9 '12 at 12:28
• Ulrich, yes, I was sloppy with my attributions. – Donu Arapura Mar 9 '12 at 12:37

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).