Frobenius splitting and cohomology vanishing for Schubert varieties

Let k be an algebraically closed field of characteristic p > 0 and X be a projective variety over k. We then have the absolute Frobenius F: X -> X and an injection Ox -* F *6x given by f fP, f e O9x. If this makes Ox a direct summand in F* Ox (as an (x-module) we call X a Frobenius split variety. For such a variety the vanishing theorem for ample line bundles follows trivially from Serre's vanishing theorem. For, tensoring Ox -F*Ox by an ample line bundle L and noting that L ? F * (x = F * F *L = F * LP (projection formula) we get that the map in cohomology H'(X, L) -+ H'(X, F*LP) = H'(X, LP) is an injection. Iterating this we see that H'(X, L) injects into H'(X, LP) for every P. But, for large v the latter is zero! Thus Frobenius split varieties have quite pleasant properties. It also turns out that using duality for the Frobenius morphism, or equivalently, the Cartier operator, one can give a manageable criterion for Frobenius splitting. The point here is the local nature of duality. The relevance of the compatibility of local and global duality was suggested to us by Grothendieck's proof of H'(X, Y) = 0 for a noncomplete variety X of dimension n ([4], Theorem 6.9) and by Kempf's paper [10]. By this criterion and the Bott-Samelson-Demazure desingularisation of Schubert varieties, it follows very easily that Schubert varieties in characteristic p are Frobenius split. The following vanishing theorem is then an immediate consequence. Let G be a reductive group over the field k (of arbitrary characteristic, zero or positive). Let Q be a parabolic subgroup and X c G/Q a Schubert variety. Let L be an ample line bundle on G/Q. Then HI(X, L) = 0 for i > 0 and the restriction map H0(G/Q, L) -+ H0(X, L) is surjective. If char k > 0 this is a consequence of the compatible Frobenius splitting of X in G/Q and the char k = 0 case is handled by semicontinuity. The above result for special Schubert varieties has been proved by SeshadriMusili-Lakshmibai [12], [13] and by Kempf [9] by using characteristic free methods. For X = G/Q, Andersen [1] and Haboush [5] have given simple proofs using characteristic p methods.