Differential operators and the Frobenius pull back of the tangent bundle on \(G/P\)
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 \textit{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. \textit{K. H. Paranjape} and \textit{V. Srinivas}, Invent. Math. 98, No. 2, 425-444 (1989; Zbl 0697.14037)] to get their result.
scientific article; zbMATH DE number 1310842