Torsion freeness of higher direct images of canonical bundles
type: Item
instance of: scholarly article
title: Torsion freeness of higher direct images of canonical bundles
zbMATH Open document ID: 0589.14016
DOI: 10.1007/BF01450836
author: Atsushi Moriwaki
published in: Mathematische Annalen
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164206
review text: Let X be a complex manifold, Y a reduced and irreducible complex space and let \(f:\quad X\to Y\) be a projective morphism. Assume that every connected component of X is mapped surjectively to Y. Then, the main purpose of this paper is to prove that \(R^ if_*\omega_ X\) is torsion free for every \(i\geq 0\), which is a local-analytic version of Kollár's theorem, where \(\omega_ X\) is the canonical bundle of X. We assume furthermore that Y is non-singular, X is equidimensional and that there is an effective divisor D on Y with only normal crossings such that f is smooth over \(Y\setminus D\). Set \({\mathcal H}^ j:=R^ jf_*({\mathbb{Z}})|_{Y\setminus D}\otimes {\mathcal O}_{Y\setminus D}\) and let \(F^ p({\mathcal H}^ j)\) be the p-th Hodge filtration of \({\mathcal H}^ j\). We set \(d=\dim X-\dim Y\). Then we can show that there are isomorphisms \(\phi_ i:\quad R^ if_*\omega_{X/Y}\simeq F^ d(^ u{\mathcal H}^{d+i})\) and \(\psi_ i:\quad R^ if_*{\mathcal O}_ X\simeq Gr^ 0_ F(^{\ell}{\mathcal H}^ i)\) for every \(i\geq 0\), where \(\omega_{X/Y}=\omega_ X\otimes f^*(\omega_ Y^{-1})\) and \({}^ u{\mathcal H}^ j\) (resp. \(^{\ell}{\mathcal H}^ j)\) is the upper (resp. lower) canonical extension of the variation of Hodge structure \({\mathcal H}^ j\). In particular, \(R^ if_*\omega_ X\) and \(R^ if_*{\mathcal O}_ X\) are locally free in this case. Moreover, we have that \(R^ if_*\omega_{X/Y}\) is semi-positive and \(R^ if_*{\mathcal O}_ X\) is semi-negative for every \(i\geq 0\) if the local monodromies are unipotent.
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 32L20
Mathematics Subject Classification ID: 14C30
Mathematics Subject Classification ID: 32L05
Mathematics Subject Classification ID: 14C20
zbMATH DE Number: 3946322
zbMATH Keywords: torsion freeness of higher direct images of canonical bundles
zbMATH Keywords: projective morphism
zbMATH Keywords: divisor with only normal crossings
zbMATH Keywords: variation of Hodge structure
MaRDI profile type: Publication
cites work: Q3663441
cites work: Higher direct images of dualizing sheaves. I
cites work: Higher direct images of dualizing sheaves. II
cites work: On 𝑛-dimensional compact complex varieties with 𝑛 algebraically independent meromorphic functions
cites work: Complex parallelisable manifolds and their small deformations
cites work: Hodge filtrations and the higher direct images of canonical sheaves
cites work: Vanishing theorems on complete Kähler manifolds
cites work: Modules de Hodge polarisables. (Polarisable Hodge modules)
cites work: Variation of Hodge structure: The singularities of the period mapping
cites work: Relative vanishing theorems in analytic spaces
cites work: On three-dimensional compact complex manifolds with non-positive Kodaira dimension
cites work: Q3658114
OpenAlex ID: W2079564822
description: scientific article; zbMATH DE number 3946322