Kodaira-Saito vanishing via Higgs bundles in positive characteristic
type: Item
instance of: scholarly article
title: Kodaira-Saito vanishing via Higgs bundles in positive characteristic
zbMATH Open document ID: 1468.14044
author: Donu Arapura
published in: Journal für die Reine und Angewandte Mathematik
publication date: 2019-10-24T00:00:00Z
full work available at URL: https://arxiv.org/abs/1611.09880
review text: Let \(X\) be a smooth projective variety defined over an algebraically closed field \(k\). Let \(D\) be a normal crossing divisor and \(L\) an ample line bundle on \(X\). Let \((E, \theta)\) be a semistable logarithmic Higgs bundle on \((X, D)\) with vanishing Chern classes in \(\ell\)-adic cohomology. If \(k\) has positive characteristic the author proves that the \(i\)-th ccohomology group of the de Rham complex of \((E, \theta)\) tensored by \(L\) is vanishing if \(i>\dim X\). Similar theorem holds also in positive characteristic if one assumes that \((X,D)\) is liftable to the ring of length \(2\) Witt vectors \(W_2(k)\). In principle, the author assumes that \(\theta\) is nilpotent, but since every semistable Higgs bundle can be deformed to a semistable one with a nilpotent Higgs field (see, e.g., Corollary 5.7 in the reviewer's paper [Doc. Math. 19, 509--540 (2014; Zbl 1330.14017)]), this assumption is not needed. The author's result implies a special case of Saito's generalization of Kodaira's vanishing theorem for mixed Hodge modules. For \(k={\mathbb C}\) the author proves also a refinement of the semipositivity theorem for complex variations of mixed Hodge structures on \((X,D)\) with unipotent local monodromies around \(D\). He also proves another variant of Saito's vanishing theorem that generalizes the Kawamata-Viehweg vanishing theorem.
reviewed by: Adrian Langer
Mathematics Subject Classification ID: 14F17
Mathematics Subject Classification ID: 14D07
Mathematics Subject Classification ID: 14G17
Mathematics Subject Classification ID: 14H60
zbMATH DE Number: 7121911
zbMATH Keywords: Higgs bundles
zbMATH Keywords: vanishing theorems
zbMATH Keywords: positive characteristic
zbMATH Keywords: complex variation of Hodge structure
zbMATH Keywords: semipositivity
MaRDI profile type: Publication
OpenAlex ID: W2963650467
arXiv ID: 1611.09880
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DOI: 10.1515/CRELLE-2017-0036
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description: scientific article