# Application of Frobenius splitting in characteristic $0$

In general, Frobenius splitting only defines on field of characteristic $p$ (algebraically closed) field.

I am reading Brion and Kumar's book and I can see that there are geometric results can be proven by it on flag variety such as all higher cohomology groups of ample line bundle vanish, rational singularity, normality etc.

I am pretty curious about what kind of results over characteristic zero were/could be proven by Frobenius splitting technique. Can anyone give me some examples or reference about this aspect?

• Techniques such as Frobenius splitting were introduced in prime characteristic because it's so hard to work out analogues involving geometry or representations of algebraic groups already known in characteristic 0. But there are fruitful interactions between characteristic $p$ questions and quantum groups (often at a root of unity) in characteristic 0, while Frobenius splitting methods have applications to quantum groups; see for instance a paper by Kumar-Littelmann, Algebraization of Frobenius splitting via quantum groups, Ann. of Math. 155 (2002), 491–551. – Jim Humphreys Nov 8 '15 at 14:54
• If you have access to MathSciNet, it may be helpful to check their list of 100+ citations of the Brion-Kumar book. But again I'd stress that the problems solved using Frobenius splitting tend to be solved already in characteristic 0 by more traditional methods. – Jim Humphreys Nov 8 '15 at 14:56
• I'm not sure this will pass even the 'techniques such as Frobenius splitting' filter, but to me two amazing applications of characteristic p to prove results in characteristic zero are Mori's proof of the Hartshorne conjecture and the Deligne-Illusie (algebraic) proof of Kodaira vanishing – aginensky Nov 8 '15 at 16:36