# Frobenius splitting

After Ben’s post on “terrible” algebraic varieties, I would like to write about some nice ones. In particular, here is a nice property for an algebraic variety X:

(*) For all ample line bundles L on X, $H^i(X, L) = 0$ for all $i > 0$.

On Tuesday, I went to a talk by Sam Payne in which he explained (among other things) how you could prove such a property using Frobenius splitting. I’d heard of this Frobenius splitting years ago from my advisor, but had no idea how it worked before.

The idea of Frobenius splitting goes back (I think) to the following paper by Mehta-Ramanathan . The idea is actually quite simple. Let X be a variety over an algebraic closed field of characteristic $p>0$. Then we have the absolute Frobenius $F : X \rightarrow X$. This is the map which is the identity on the underlying topological space and acts by $f \mapsto f^p$ on the sheaf of functions. So we have an algebra map $O_X \rightarrow F_* O_X = O_X$

Now a Frobenius splitting is a splitting of this map as a map of $O_X$ modules. Here is the amazing result:

A Frobenius splitting gives the property (*).

The proof of this is pretty simple, but rather than give it here, I’ll just refer you to the first page of the above paper.

The main example of a Frobenius split variety is a flag variety of any type. Even Schubert varieties in arbitrary flag varieties are Frobenius split. One other interesting thing is that though the notion of Frobenius split applies just to varieties in characteristic p, it can be used to prove (*) in characteristic 0 by a “semicontinuity” principle which I’ve never quite understood (perhaps one of our astute readers can explain it).

Another nice thing about Frobenius splitting is that to achieve the same results, you don’t actually need the Frobenius map. For example, Payne explained that for toric varieties over any characteristic, there is a map corresponding to F which is just $t \rightarrow t^m$. This map can be split in an analogous manner and doing so gives (*) for toric varieties. For flag varieties, there is also the approach of Littelmann and Kumar which uses the Lusztig’s Frobenius map on quantum groups.

## 11 thoughts on “Frobenius splitting”

1. here’s how semi-continuity works in this case:

take your favorite variety $X$ proper over the integers (probably you can get away with less than proper. David probably knows how), and your favorite line bundle $L$ on $X$.

Now, as we all know, there is a canonical map $\phi:X\to\mathrm{Spec}\,\mathbb{Z}$. Consider the higher pushforwards $R^i\phi_*L$. This is a sheaf on $\mathrm{Spec}\,\mathbb{Z}$, and by proper base change, the pullback of this sheaf to any point of $\mathrm{Spec}\,\mathbb{Z}$ is the $i$-th cohomology of $L$ pulled back to the fiber.

If we pick a point corresponding to a prime, we’ll get the cohomology of our variety over $\mathbb{F}_p$, and over the generic point we’ll get the cohomology for the same variety of $\mathbb{Q}$. If the cohomology vanishes over $\mathbb{F}_p$, then by Nakayama’s lemma, the sheaf $R^i\phi_*L$ has trivial stalk at that point. The stalk of any sheaf at the generic point has lower or equal rank to its stalk at any closed point, so the cohomology vanishes in characteristic 0.

2. Jason Starr says:

Frobenius splitting also has consequences for the singularities of a variety. I believe this has been worked out by many people (I have heard Karen Smith speak about singularities and F-splitting and F-regularity). But I’ve mostly heard about this from my colleague Xuhua He who uses Frobenius splitting to study Lusztig’s “G-stable pieces” in the wonderful compactification of an algebraic group of adjoint type.

3. bb says:

You have to be careful about cohomology and base change! The formation of higher direct images does *not* commute with base change, even for proper smooth maps and vector bundle coeffecients. For example, the plurigenera (the dimensions of H^0(X,mK_X)) can go up when you specialise from char 0 to char p (see Junecue Suh’s 2007 princeton thesis for some explicit examples coming from Shimura varieties). What does commute with *derived* base change is the formation of the derived pushforwards (of flat sheaves under proper maps).

The consequence you wanted (vanishing mod p implies vanishing in char 0 under suitable hypothesis) can be obtained from Zariski’s formal functions theorem (which says that cohomology of the generic fibre can be computed using cohomology of the (formal completion of the) special fibre, for a proper family over a complete noetherian local ring).

4. Allen Knutson says:

1) An actual Frobenius splitting, rather than merely the knowledge that one exists, can be extremely useful. Consider it as a sort of geometric object one might have on a space, like a spin structure or a foliation or something. In particular, the concept of “compatible Frobenius splitting”, where a splitting of X is sort of tangent to a subscheme Y, is really handy: being split implies being reduced, and the intersection of compatibly split subschemes (w.r.t. the same splitting) is again compatibly split, so one can use this to show that some intersections are reduced.

2) I would like to see some definitive statement relating being Frobenius split and some sort of positivity. In particular, being Fano means that one has nice enough sections of the anticanonical to build a Frobenius splitting, and then the zero set is compatibly split.

(On the flag manifold that Joel mentions, one can get this zero set to be the union of the Schubert and the antiSchubert divisors. Alternate taking components and intersecting, and one learns that every Richardson variety is compatibly split, hence one has the nice vanishing.)

When I’ve wanted to prove vanishing using Frobenius splitting, I’ve generally found this positivity to be what’s really available/necessary, and can then use Kawamata-Viehweg instead (not that it’s easier, it’s just characteristic 0).

5. Ooph. I am revealed yet again to be faking it as an algebraic geometer. Remind me to just stay in the derived category next time.

I would maintain that it is satisfying as a heuristic story of why vanishing over characteristic $p$ should imply vanishing over characteristic 0. But of course, we should be careful about what is a proof and what is a bedtime story.

6. Jesse Kass says:

bb,
Nobody has stated this explicitly, but the line bundle L needs to be flat over Z for the result to be true as stated.

The result also following immediately from the Semicontinuity Theorem. The Semicontinuity Theorem says that under your conditions (X –> Spec(Z) is proper, L = Z-flat line bundle), then the dimension of the cohomology of a fiber can only increase under specialization.

In the current set-up, the dimension of the cohomology of the generic fiber can be at most the dimension of the cohomology of the fiber at p (which is equal to zero).

Also, the Semicontinuity Theorem combined with Grauert’s Theorem implies that the formation of higher direct images commutes with taking the stalk at a _general_ point in the base.

p.s. b.b., are you Bhargav?

7. bb says:

jesse, i said flat for the derived base change theorem (which gives semicontinuity). [i don’t think you didn’t need any flatness assumptions for the formal functions theorem (ega3.1, thm 4.1.5).]

and yes, i’m bhargav.

8. Jesse Kass says:

Wow, Bhargav! Fancy meeting you here! How’s it going?

The Theorem on Formal Functions does not require flatness. What I meant is that the statement that H^{i}(X|_{p},L|_{p})=0 implies that H^{i}(X_{Q},L_{Q})=0 requires L to be flat over Z (I think).

If I remember correctly, the flatness assumption comes up in the proof as follows. Assume that H^{i}(X|_{p},L|_{p})=0. To apply the theorem on formal function, I need to show that

lim H^{i}(X_{p^n},L|_{p^n}) = 0

Here X|_{p^n} is the n-th infinitesimal neighborhood of X|_{p} (sorry for the awful notation). The limit is an inverse limit. To prove this, one proves that in fact H^{i}(X_{p^n},L|_{p^n}) =0 for all n. The proof of this uses the flatness of L over Z.

best,
-jk

9. Anonymous says:

2) I would like to see some definitive statement relating being Frobenius split and some sort of positivity. In particular, being Fano means that one has nice enough sections of the anticanonical to build a Frobenius splitting, and then the zero set is compatibly split.

Sorry, are you stating a fact or a conjecture?

10. Allen Knutson says:

Sorry, are you stating a fact or a conjecture?

I was stating a fact (Fanos are split for p>>0; see the book [Brion-Kumar] exercises 1.6.E), and I was also conjecturing that there should be some sort of converse, but not suggesting what it might be.