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Differential operators and the Frobenius pull back of the tangent bundle on \(G/P\). (English) Zbl 0935.14032

The main result of this paper is the following theorem: Let \(G\) be an almost simple, simply connected algebraic group over an algebraically closed field of positive characteristic \(p\). Let \(P\) be a proper parabolic subgroup of \(G\), put \(X:=G/P\) and let \(F\colon X\to X'\) be the Frobenius morphism on \(X\). If \(H^1(X,F^*T_{X'})=0\) (here \(T_{X'}\) denotes the tangent bundle on \(X')\), then \(X\cong {\mathbb P}^n\) for some \(n>0\). This is a generalization of a result of B. Haastert [cf. Manuscr. Math. 58, 385-415 (1987; Zbl 0607.14010)]. The present authors use deformation theory and the “complex geometry” of \(X\) [cf. K. H. Paranjape and V. Srinivas, Invent. Math. 98, No. 2, 425-444 (1989; Zbl 0697.14037)] to get their result.

MSC:

14M17 Homogeneous spaces and generalizations
14G15 Finite ground fields in algebraic geometry
14D15 Formal methods and deformations in algebraic geometry
13A35 Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure
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