Positivity on subvarieties and vanishing of higher cohomology
type: Item
instance of: scholarly article
title: Positivity on subvarieties and vanishing of higher cohomology
zbMATH Open document ID: 1291.14018
author: Alex Küronya
publication date: 2014-04-14T00:00:00Z
full work available at URL: https://arxiv.org/abs/1012.1102
review text: The article under review studies the relationship between the line bundle with partial positivity and vanishing of their higher cohomology. Let \(L\) be a Cartier divisor \(L\) over a complex projective scheme \(X\). If the restriction of \(L\) on \(E_1 \cap\cdots \cap E_q\) is ample, where \(E_i\) is a general element of \(|A_j|\) for some very ample Cartier divisor \(A_j\) on \(X\), the author proved the following vanishing theorem: \[ H^i (X, \mathcal{F}\otimes \mathcal{O}_X (mL +N))=0 \qquad\text{for all } i>q , \] where \(\mathcal{F}\) is an arbitrary coherent sheaf on \(X\), \(m > m(L, \mathcal{F})\) and \(N\) is an arbitrary nef divisor. In the case \(X\) is smooth and the restriction of \(L\) on \(E_1 \cap\cdots \cap E_q\) is big and nef, the author also proved a Kawamata-Viehweg type vanishing theorem: \[ H^i (X, K_X\otimes \mathcal{O}_X (L))=0 \qquad\text{for all } i>q . \] The proofs of these vanishing theorems uses a standard complex and some induction arguments. Thus the proof is purely algebraic. Compare to many other vanishing theorems, a remarkable feature of this article is that the divisor \(L\) here is not necessarily pseudo-effective.
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 14F17
zbMATH DE Number: 6284530
zbMATH Keywords: vanishing of higher cohomology
zbMATH Keywords: partial positivity
MaRDI profile type: Publication
OpenAlex ID: W2963729197
arXiv ID: 1012.1102
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DOI: 10.5802/AIF.2812
published in: Annales de l’institut Fourier
description: scientific article