Positivity of direct image sheaves. A geometric point of view
type: Item
instance of: scholarly article
title: Positivity of direct image sheaves. A geometric point of view
zbMATH Open document ID: 1203.14011
DOI: 10.4171/LEM/56-1-4
author: Andreas Höring
published in: L'Enseignement Mathématique. 2e Série
publication date: 2010-09-20T00:00:00Z
review text: Suppose that \(\varphi : X \to Y\) is a fibration (a morphism with connected fibers) between two complex projective manifolds. Over the last 30 years, it has been incredibly important to understand the positivity properties of \(\varphi_* \omega_{X/Y}^{\otimes n}\) or more generally of \(\varphi_*( \omega_{X/Y} \otimes L)\) where \(L\) is a line bundle which is also in some sense positive. The goal of collection of lecture notes is to explain, with many examples, the connections between these positivity properties and the geometry of \(\varphi\). The history and a description of some of the many and varied applications of these ideas are laid out in the introduction. In the second section, a series of examples is given showing what can (and cannot) happen. In the third section, some of the ideas of the proofs of these positivity theorems are explained (including variants for higher direct images). Finally, in section 4, these ideas are applied to the study of projective manifolds with nef anti-canonical line bundle. An appendix on reflexive sheaves, singularities and duality is also included.
reviewed by: Karl Schwede
Mathematics Subject Classification ID: 14D06
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 14F17
Mathematics Subject Classification ID: 14F18
Mathematics Subject Classification ID: 14E30
zbMATH DE Number: 5788665
zbMATH Keywords: direct image
zbMATH Keywords: adjoint line bundle
zbMATH Keywords: fibration
zbMATH Keywords: multiplier ideal
zbMATH Keywords: vanishing theorem
zbMATH Keywords: extension theorem
zbMATH Keywords: canonical bundle
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4171/lem/56-1-4
OpenAlex ID: W2003089702
description: scientific article; zbMATH DE number 5788665