Projective normality of flag varieties and Schubert varieties.

*(English)*Zbl 0553.14023We prove in this paper the following results. Let G be a semisimple algebraic group over an algebraically closed field k and Q a parabolic subgroup containing a Borel subgroup B. Let X be a Schubert variety (i.e. the closure of a B orbit) in G/Q. Then (a) If L is a line bundle on G/Q such that \(H^ 0(G/Q,L)\neq 0\) then \(H^ i(X,L)=0\) for \(i>0\) and the restriction map \(H^ 0(G/Q,L)\to H^ 0(X,L)\) is surjective; (b) X is normal; (c) X is projectively normal in any embedding given by an ample line bundle on G/Q. - If we prove the results for fields of positive characteristic they follow for fields of characteristic zero by semicontinuity. When char k\(=0\) we have the absolute Frobenius morphism \(F:X\to X\) defined by raising functions on X to the p-th power. In the preprint ”Frobenius splitting and cohomology vanishing for Schubert varieties” by V. B. Mehta and A. Ramanathan it was shown using duality for the Frobenius morphism of the Bott-Samelson-Demazure variety (constructed in the paper of Demazure cited below) that the p-th power map \(0_ X\to F_*0_ X\) admits a section. This quickly gives (a) for ample line bundles L. In this paper we extend this method, by a closer examination of the splitting, to the general case of \(H^ 0(G/Q,L)\neq 0\). We then deduce (b) from (a) by an inductive argument involving the \({\mathbb{P}}^ 1\)-fibrations \(G/B\to G/P\) for suitable minimal parabolic subgroups P containing B.

These results prove the conjectures of M. Demazure in his paper in Ann. Sci. Éc. Norm. Supér., IV. Sér. 7, 53-88 (1974; Zbl 0312.14009). In particular his character formula for \(H^ 0(X,L)\) for fields of arbitrary characteristic also follows. Incidentally our results uphold the main claims in Demazure’s paper in spite of the falsity of proposition 11, §2 of that paper.

These results prove the conjectures of M. Demazure in his paper in Ann. Sci. Éc. Norm. Supér., IV. Sér. 7, 53-88 (1974; Zbl 0312.14009). In particular his character formula for \(H^ 0(X,L)\) for fields of arbitrary characteristic also follows. Incidentally our results uphold the main claims in Demazure’s paper in spite of the falsity of proposition 11, §2 of that paper.

##### MSC:

14M15 | Grassmannians, Schubert varieties, flag manifolds |

14L30 | Group actions on varieties or schemes (quotients) |

##### Keywords:

projective normality; vanishing cohomology groups; Schubert variety; line bundle; Frobenius morphism##### Citations:

Zbl 0312.14009
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\textit{S. Ramanan} and \textit{A. Ramanathan}, Invent. Math. 79, 217--224 (1985; Zbl 0553.14023)

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##### References:

[1] | Andersen, H.H.: The Frobenius morphism on the cohomology of homogeneous vector bundles onG/B. Ann. Math.112, 113-121 (1980) · Zbl 0437.20035 |

[2] | Demazure, M.: Désingularisations des variétés de Schubert généralisés, Ann. E.N.S.7, 53-88 (1974) · Zbl 0312.14009 |

[3] | Kempf, G.: Linear systems on homogeneous spaces. Ann. Math.103, 557-591 (1976) · Zbl 0327.14016 |

[4] | Mehta, V.B., Ramanathan, A.: Frobenius splitting and cohomology vanishing for Schubert varieties. · Zbl 0601.14043 |

[5] | Ramanathan, A.: Schubert varieties are arithmetically Cohen-Macaulay. (To appear) · Zbl 0541.14039 |

[6] | Seshadri, C.S.: Line bundles on Schubert varieties. (To appear) · Zbl 0688.14047 |

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