Weak Akizuki-Nakano vanishing theorem for singular globally \(F\)-split 3-folds
type: Item
instance of: scholarly article
title: Weak Akizuki-Nakano vanishing theorem for singular globally \(F\)-split 3-folds
zbMATH Open document ID: 1544.1301
author: Kenta Sato
author: Shunsuke Takagi
published in: Manuscripta Mathematica
publication date: 2024-07-01T00:00:00Z
review text: It is well-known that the (Kodaira-)Akizuki-Nakano vanishing theorem fails in positive characteristic, but under additional hypotheses, such as being smooth and globally \(F\)-split, it may hold.\N\NThe present paper proves the following version of KAN vanishing: Let \(X\) be a projective globally \(F\)-regular variety over a perfect field of characteristic \(p > 0\) and \(L\) be a globally generated ample line bundle on \(X\) such that the map defined by \(L\) is generically étale onto its image. If \(\dim X \le 3\) and \(X\) has only isolated singularities, then \(X\) satisfies KAN vanishing (for the sheaves of reflexive differentials \(\Omega_X^{[p]}\)).\N\NThere is also a version for the sheaves of Kähler differentials \(\Omega_X^{p}\), under stronger hypotheses. Two applications are given: one to the deformation theory of globally \(F\)-split Fano threefolds and the other to Kodaira vanishing on infinitesimal thickenings of lci subvarieties of projective space. The latter recovers a result by \textit{B. Bhatt} et al. [Am. J. Math. 141, No. 2, 531--561 (2019; Zbl 1425.13008)].
reviewed by: Patrick Graf
Mathematics Subject Classification ID: 13A35
Mathematics Subject Classification ID: 14F17
Mathematics Subject Classification ID: 14J45
Mathematics Subject Classification ID: 14D15
zbMATH DE Number: 7873597
zbMATH Keywords: Akizuki-Nakano vanishing
zbMATH Keywords: globally F-split
zbMATH Keywords: singular threefolds
zbMATH Keywords: deformation theory
zbMATH Keywords: infinitesimal thickenings
MaRDI profile type: Publication
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DOI: 10.1007/S00229-023-01529-9
description: scientific article; zbMATH DE number 7873597