The Frobenius morphism on a toric variety
type: Item
instance of: scholarly article
title: The Frobenius morphism on a toric variety
zbMATH Open document ID: 0899.14026
author: Jesper F. Thomsen
author: Niels Lauritzen
author: Anders Skovsted Buch
author: Vikram B. Mehta
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1998-11-11T00:00:00Z
review text: 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 \(\mathbb{Z}/p^2 \mathbb{Z}\).
Mathematics Subject Classification ID: 14M25
Mathematics Subject Classification ID: 14F17
Mathematics Subject Classification ID: 14M15
Mathematics Subject Classification ID: 13A35
zbMATH DE Number: 1140475
zbMATH Keywords: Frobenius morphism
zbMATH Keywords: characteristic \(p\)
zbMATH Keywords: characteristic \(p^2\)
zbMATH Keywords: Bott vanishing theorem
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178225109
OpenAlex ID: W1970971420
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DOI: 10.2748/TMJ/1178225109
description: scientific article; zbMATH DE number 1140475