A note on the dimension of global sections of adjoint bundles for polarized 4-folds
type: Item
instance of: scholarly article
title: A note on the dimension of global sections of adjoint bundles for polarized 4-folds
zbMATH Open document ID: 1405.14021
author: Yoshiaki Fukuma
published in: Proceedings of the Japan Academy. Series A
publication date: 2018-09-21T00:00:00Z
review text: Let \((X,L)\) be a polarized manifold of dimension \(n\) over the complex field, where \(L\) is ample. The author of the article under review is interested in establishing a lower bound on \(h^0(K_X+tL)\) for \(t>0\). To this end, in a previous paper he introduced an invariant \(A_i(X,L)\), for any \(0\leq i\leq n\), called \(i\)-th Hilbert coefficient of \((X,L)\). For a fixed \(t\), it is defined as follows. First define inductively \(F_0(t)=h^0(K_X+tL)\), and \(F_i(t)=F_{i-1}(t+1)-F_{i-1}(t)\). Then define \(A_i(X,L)=F_{n-i}(1)\). Here the case \(n=4\) is considered. It is proved that \(A_2(X,L)\geq 0\). Moreover, if \(m\geq 3\) it is proved that \(h^0(K_X+mL)\geq h^0(K_X+(m-1)L)\). A geometrical description is also given of the \(4\)-folds possibly failing the inequality in the case \(m=2\). This answers to a question posed by H. Tsuji.
reviewed by: Emilia Mezzetti
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 14J35
zbMATH DE Number: 6941822
zbMATH Keywords: polarized manifold
zbMATH Keywords: adjoint bundle
zbMATH Keywords: sectional geometric genus
MaRDI profile type: Publication
full work available at URL: pjaa.94.53
OpenAlex ID: W2799952160
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DOI: 10.3792/PJAA.94.53
description: scientific article; zbMATH DE number 6941822