Frobenius splitting and cohomology vanishing for Schubert varieties.

*(English)*Zbl 0601.14043Let \(F\) 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 V. Lakshmibai, C. Musili and 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 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 O. Mathieu [C. R. Acad. Sci., Paris, Ser. I 303, 391–394 (1986; Zbl 0602.17008)] and S. Kumar [”Demazure character formula in arbitrary Kac-Moody setting”, preprint (Tata Inst. 1986); Invent. Math. 89, 395–423 (1987; Zbl 0635.14023)].

The second author has followed up on these results by proving – in joint work with 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 O. Mathieu [C. R. Acad. Sci., Paris, Ser. I 303, 391–394 (1986; Zbl 0602.17008)] and S. Kumar [”Demazure character formula in arbitrary Kac-Moody setting”, preprint (Tata Inst. 1986); Invent. Math. 89, 395–423 (1987; Zbl 0635.14023)].

Reviewer: H. H. Andersen (Aarhus)

##### MSC:

14M15 | Grassmannians, Schubert varieties, flag manifolds |

14G15 | Finite ground fields in algebraic geometry |

20G10 | Cohomology theory for linear algebraic groups |