On a conjecture of Beltrametti-Sommese for polarized 4-folds
type: Item
instance of: scholarly article
title: On a conjecture of Beltrametti-Sommese for polarized 4-folds
zbMATH Open document ID: 1327.14038
author: Yoshiaki Fukuma
published in: Kodai Mathematical Journal
publication date: 2015-09-11T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.kmj/1436403895
review text: Let \((X,L)\) be a polarized manifold of dimension \(n\), i.e., \(X\) is a smooth complex projective variety and \(L\) is an ample line bundle on \(X\). \textit{M. C. Beltrametti} and \textit{A. J. Sommese} conjectured in Conjecture 7.2.7 on [The adjunction theory of complex projective varieties. De Gruyter Expositions in Mathematics. 16. Berlin: de Gruyter (1995; Zbl 0845.14003)] that for \(n \geq 3\) the nefness of the adjoint bundle \(K_X+(n-1)L\) implies its effectivity, that is \(h^0(X, K_X+(n-1)L)>0\). The conjecture is proved to be true when \(n=3\) and when \(L\) is effective (see the Introduction of the paper and references therein). The main result of the paper is to prove the conjecture when \(n=4\) (see Thm. 3.1). The proof is a case by case analysis where the maximal rationally connected fibration of \(X\) plays an important role in one of the cases (\(q(X)=0\) and \(\Omega_X\langle L \rangle\) not generically nef).
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 14J35
zbMATH DE Number: 6481139
zbMATH Keywords: polarized manifolds
zbMATH Keywords: adjoint bundles
zbMATH Keywords: nef and effective divisors
zbMATH Keywords: sectional genus
reviewed by: Roberto Muñoz
MaRDI profile type: Publication
Wikidata QID: Q123111416
OpenAlex ID: W1524517030
DOI: 10.2996/KMJ/1436403895
description: scientific article; zbMATH DE number 6481139