On the dimension of global sections of adjoint bundles for polarized 3-folds and 4-folds
type: Item
instance of: scholarly article
title: On the dimension of global sections of adjoint bundles for polarized 3-folds and 4-folds
zbMATH Open document ID: 1126.14009
author: Yoshiaki Fukuma
published in: Journal of Pure and Applied Algebra
publication date: 2007-09-03T00:00:00Z
review text: Let \((X,L)\) be a polarized manifold, i.e., \(X\) is a smooth complex projective variety and \(L\) is an ample line bundle on \(X\). Denote by \(n\) the dimension of \(X\). It is of interest to study the properties of the adjoint bundles \(K_X+mL\) (\(K_X\) stands for the canonical bundle of \(X\) and \(m\) is an integer) of \((X,L)\). In particular it is a natural question to know when the adjoint linear systems \(| K_X+mL| \) are non empty. The paper under review gives some answers to this question when \(n=3\) and \(n=4\). In particular, when \(n=3\) an explicit lower bound (\(>0\)) for \(h^0(X,K_X+L)\) is provided when the Kodaira dimension of \(X\) is non-negative. Moreover, under the assumption that the Iitaka dimension of \(K_X+L\) is non-negative, it is shown that the emptiness of \(| K_X+L| \) implies not only negative Kodaira dimension of \(X\) but \(h^1({\mathcal O}_X)=0\). When \(n=4\), for \(X\) of non-negative Kodaira dimension, a lower bound (also \(>0\)) is given for \(h^0(X,K_X+mL)\), \(m \geq 2\). If moreover \(h^1({\mathcal O}_X)>0\) then \(| K_X+L| \) is non-empty. The results of the paper provide partial answers to a famous conjecture of Beltrametti and Sommese which states that for \(n \geq 3\) the nefness of \(K_X+(n-1)L\) implies the non-emptiness of \(| K_X+(n-1)L| \). In particular, for \(n=3,4\) the conjecture is true unless \(h^1({\mathcal O}_X)=0\) and Kodaira dimension of \(X\) equal to \(-\infty\).
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 14C17
Mathematics Subject Classification ID: 14C40
Mathematics Subject Classification ID: 14J40
zbMATH DE Number: 5186651
zbMATH Keywords: adjoint bundles
reviewed by: Roberto Muñoz
MaRDI profile type: Publication
full work available at URL: j.jpaa.2007.02.010
OpenAlex ID: W2015065739
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DOI: 10.1016/J.JPAA.2007.02.010
description: scientific article; zbMATH DE number 5186651