In this post we prove that varieties over characteristic fields are all ‘Hodge symmetric’, meaning that .
We call a smooth, projective, integral variety Hodge symmetric (or say that it satisfies Hodge symmetry), just in case . Recall the definition of the Hodge numbers . The notion of Hodge symmetry is most likely first encountered in the context of complex manifolds , where a similar form of Hodge symmetry holds:
It turns out that this actually holds, more generally, over any field of characteristic , and in fact, the complex geometry version is the deciding reason why it does hold. The key is that, by using general functoriality, and a healthy dose of GAGA, we can reduce this to the complex analytic case, a technique which should be widely applicable.
Remark: I would like to say that this is the only proof of ‘Hodge symmetry holds in characteristic ‘ (a widely made claim) that I can think of. That said, I have a sneaking suspicion that there is a more elementary way to attack this problem. If you can point out the silly argument that I am missing, please do!
Reduction to complex case
Let’s rigorously state our theorem:
Theorem(Main): Let be a characteristic field. Then, any smooth, projective, integral variety satisfies Hodge symmetry.
The somewhat non-obvious part to the proof of this, is that we can reduce the proof to the case when . More specifically:
Theorem(Reduction to ): The main theorem is true if and only if it holds for .
Lemma: Let and be two algebraically closed fields of the same characteristic, and the same uncountable cardinality. Then, .
Proof: Let the prime subfields of and be . Take transcendence bases for and for . Note that since is algebraic, and is algebraically closed, it must be the case that . Similarly, . Now, since is countable, and and uncountable, a simple cardinality argument shows that and must be equipotent. So, , and thus
Proof of Reduction to : Let be an arbitrary characteristic field, and let be a smooth, projective, integral variety. Choose an embedding . We know then that for some homogenous polynomials . Suppose that
where is a tuple in , and . Let .
Now, note that if we let be the obvious map, then . Moreover, it follows then that
and so, in particular, for all . Thus, to prove that , it suffices to prove such a statement for .
Now, let . Note that since is countable, that , and so by the lemma, . Thus, we obtain an embedding of fields . Let . Note that the same argument as before shows that for all . Since is obviously smooth, projective, and integral, the claim follows.
It’s clear that one could imitate the above proof if many situations. Most specifically, let us informally say that a collection of quasicoherent sheaves , as ranges over all characteristic fields, and ranges over all smooth integral varieties over (one could take any subcategory of varieties over ), is functorial if where is a field extension, and is the obvious map.
Then, there is an easy to generalize meta-theorem lurking the above reduction to . If we have two functorial collections of sheaves and we call a relation between and functorial if for all extensions of fields the relationship between and holds if and only if the relationship between and holds. Then, the meta-theorem should be as follows:
Meta-theorem: The relationship between and should hold for all and all , if and only if it holds for all when .
For example, the collection of quasicoherent sheaves and is functorial, and the relationship that is functorial. So, the meta-theorem implies the proof of the reduction to theorem. Of course, the proof of this meta-theorem is also just the same as in the proof of the reduction to theorem.
This is a very powerful meta-theorem since if the and are coherent, then by reducing to , and applying the GAGA principle we can reduce these theorems to an analytic setting, where we might have more tools.
I think the above is, in some sense, just a disguised form of another well-known meta-theorem. Namely, that of the Lefschetz principle. This roughly says that all statements about varieties over algebraically characteristic fields , which are first-order definable in the language of fields, hold for general if and only if they hold for . This is just a teased out statement that the theory of algebraically closed fields of characteristic is complete.
This is further cemented by the fact that the proof that is complete uses the following elementary result from model theory:
Theorem(Łoś–Vaught test): Let be a (first-order) theory with no finite models, and which is -categorical for some uncountable cardinal . Then, is complete.
Note then that the lemma above is just checking that is -categorical for all uncountable ! Since obviously has no finite models, the Łoś–Vaught test tells us that is complete.
Despite the obvious similarities between our meta-theorem and the Lefschetz principle, it’s not clear to me how they precisely relate.
Intuitively one would expect that any functorial relationship between two functorial collections of sheaves is first-order definable. This is because independence of the base field should make you think that the statement can be coded using only the basic operations of the fields (i.e. in the first order language of fields). If this were true, then our meta-theorem really would be a special case of the Lefschetz principle. But, I don’t see how to show that such functorial relationships are first-order in general.
The complex case
So, now that we have the reduction to out of the way, we can focus our attention entirely on the complex case.
Now, the following theorem is fairly elementary, and while a much, much, much easier special case, is still an instance of the above-cited GAGA principle.
Theorem(Complex Hodge Symmetry): Let be a smooth, projective, integral variety. Then,
where is the analytification of , and denotes holomorphic -forms.
So, we have reduced the case over to showing that if is a compact Kahler manifold (which all analytifications of smooth projective integral varieties are), then
But, in case you haven’t noticed yet, our post has been a bit thematic so-far. Namely, this is the post of reductions, and after that, more reductions. Thus, you shouldn’t be surprised that we are not done with the reductions. Namely, we reduce this identity to a statement about Dolbeault cohomology.
Let us quickly recall the definition of Dolbeault cohomology on a complex manifold . Let is fix , and note that we have the following chain complex of sheaves
Here is the sheaf of -forms on which, in local (complex) coordinates are the -span of forms of the type
The map is just the map defined in local coordinates by
It’s easy to check that , and so we really do have a chain complex.
We then define the Dolbeault cohomology of , denoted , to be the cohomology of
at the group.
Just as in the case of de Rham cohomology, we can relate the Dolbeault cohomology to the Hodge groups (notice the switch in indexing!) by realizing that the Dolbeault sequence is a resolution of the sheaf . More explictly
Theorem(Dolbeault’s theorem): For all pairs , we have an isomorphism, and any complex manifold
Proof: Note that we have the Dolbeault sequence
Before we justify this, note the difference between and . In local coordinates, both look something like
but in , we have that must be holomorphic, whereas in , is just a smooth -valued function. The map is just inclusion.
Now, let’s prove exactness of this sequence. Clearly the first map is injective. The fact that is the kernel of reduces to the fact that a smooth function is holomorphic if and only if for . The exactness at the other points reduces to the so-called Dolbeault lemma, which is an exercise in tedium, very much similar to the similar de Rham theorem.
Now, note that since each is a -module ( being the sheaf of smooth -valued functions on ), it is fine, and so acyclic. Thus, since we can compute the cohomology of a sheaf from an acyclic resolution, we see that is the -cohomology of the sequence
but, this is precisely the group .
So, why is Dolbeault’s theorem helpful in proving complex Hodge symmetry? Well, because then we may apply a well-known (but difficult!) theorem:
Theorem(Hodge symmetry for Kahler manifolds): Let be a compact Kahler manifold, then
The proof of this result requires too much background to discuss here (a rough sketch of the proof, and its importance are discussed in this post of mine). Roughly the idea is that one takes the Kahler metric on , and then uses this to define the notion of harmonic -forms with respect to this metric. The set of harmonic -forms are conjugate to the group of harmonic -forms, and so the groups are obviously isomorphic. One then checks, that can be identified with the harmonic -forms.