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, for all .
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 . Then we have the absolute Frobenius . This is the map which is the identity on the underlying topological space and acts by on the sheaf of functions. So we have an algebra map
Now a Frobenius splitting is a splitting of this map as a map of 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 . 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.