The Frobenius morphism on a toric variety
Let \(X\) be a projective toric variety over a field \(k\). In Russ. Math. Surv. 33, No. 2, 97-154 (1978); translation from Usp. Mat. Nauk 33, No. 2(200), 85-134 (1978; Zbl 0425.14013), \textit{V. I. Danilov} states the Bott vanishing theorem \(H^i(X, \widetilde \Omega^j_{ X/k} \otimes L)=0\) where \(\widetilde \Omega^j_{X/k}\) denotes the Zariski differentials, \(L\) is an ample line bundle on \(X\) and \(i>0\). \textit{V. V. Batyrev} and \textit{D. A. Cox} proved this theorem in the simplicial case in Duke Math. J. 75, No. 2, 293-338 (1994; Zbl 0851.14021). We give a characteristic \(p\) proof of the Bott vanishing theorem for projective toric varieties using that the Frobenius morphism on a toric variety lifts to characteristic \(p^2\). In the second half of this paper we study the Frobenius morphism on flag varieties. This is related to the work of \textit{K. H. Paranjape} and \textit{V. Srinivas} [Invent. Math. 98, No. 2, 425-444 (1989; Zbl 0697.14037)]. They have proved using complex algebraic geometry that if Frobenius for a flag variety \(X\) over \(k\) lifts to the \(p\)-adic numbers \(\mathbb{Z}_p= \text{proj lim}_n \mathbb{Z}/p^n \mathbb{Z}\), then \(X\) is a product of projective spaces. We generalize this result by showing that Frobenius for a large class of flag varieties admits no lift to \(\mathbb{Z}/p^2 \mathbb{Z}\).
scientific article; zbMATH DE number 1140475
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