Frobenius splitting and cohomology vanishing for Schubert varieties
Let \(F\) denote the Frobenius endomorphism on a projective variety \(X\) over a field of positive characteristic. Then \(X\) is called Frobenius split if the map \(\mathcal O_ X\to F_* \mathcal O_X\) splits. The main result in this paper says that Schubert varieties in \(G/B\), \(G\) a connected reductive group with Borel subgroup \(B\), are all Frobenius split. As a consequence the authors obtain the following vanishing theorem: If \({\mathcal L}\) is an ample line bundle on a Schubert variety \(X\), then \(H^i(X,\mathcal L) = 0\) for \(i>0\). In the case \(X=G/B\) this is a special case of Kempf's vanishing theorem. For certain classes of Schubert varieties it was established by \textit{V. Lakshmibai}, \textit{C. Musili} and \textit{C. Seshadri}, see e.g. [Bull. Am. Math. Soc., New. Ser. 1, 432--435 (1979; Zbl 0466.14020)]. The second author has followed up on these results by proving -- in joint work with \textit{S. Ramanan} -- that Schubert varieties are normal [Invent. Math. 79, 217--224 (1985; Zbl 0553.14023)] and Cohen-Macaulay [Invent. Math. 80, 283--294 (1985; Zbl 0541.14039)]. The normality, a more general vanishing theorem and, as a consequence, Demazure's character formula were also proved by the reviewer [Invent. Math. 79, 611--618 (1985; Zbl 0591.14036)]. Generalizations to the Kac-Moody case have been obtained by \textit{O. Mathieu} [C. R. Acad. Sci., Paris, Ser. I 303, 391--394 (1986; Zbl 0602.17008)] and \textit{S. Kumar} [''Demazure character formula in arbitrary Kac-Moody setting'', preprint (Tata Inst. 1986); Invent. Math. 89, 395--423 (1987; Zbl 0635.14023)].
scientific article; zbMATH DE number 3968783