The Mordell-Bombieri-Noguchi conjecture over function fields
type: Item
instance of: scholarly article
title: The Mordell-Bombieri-Noguchi conjecture over function fields
zbMATH Open document ID: 0653.14017
author: Kazuhisa Maehara
published in: Kodai Mathematical Journal
publication date: 1988-01-01T00:00:00Z
review text: The author discusses the higher dimensional analogue of Mordell's finiteness conjecture over function fields. The algebraic fibre space whose general fibre a variety of general type is conjectured to be birationally trivial, i.e. birationally equivalent to the product of the base space and a general fibre, if there are an infinite number of cross sections whose union is Zariski dense. Many authors have proved this conjecture under some additional conditions. - In this paper the author proves it supposing that \({\mathcal O}(\alpha)\otimes p^*{\mathcal O}(-K_ X)\) is \(f\circ p\)-nef for some \(\alpha >0\), where p is the projection, \(p(\Omega^ s_ X)\to X\), where \(s=\dim (S)\) and \(f:\quad X\to S\) is a proper surjective map. - The main theorem follows from the authors previous result [see Math. Ann. 262, 101-123 (1983; Zbl 0438.14011)] and by a vanishing theorem due to \textit{J. Kollár} [Ann. Math., II. Ser. 123, 11-42 (1986; Zbl 0598.14015) and 124, 171-202 (1986; Zbl 0605.14014)].
Mathematics Subject Classification ID: 14J15
Mathematics Subject Classification ID: 14E05
Mathematics Subject Classification ID: 14H05
zbMATH DE Number: 4065162
zbMATH Keywords: numerical effective bundle
zbMATH Keywords: higher dimensional analogue of Mordell's finiteness conjecture over function fields
zbMATH Keywords: nef
reviewed by: Hironobu Maeda
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138038811
OpenAlex ID: W1983689741
Wikidata QID: Q123190210
cites work: Arakelov's theorem for abelian varieties
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cites work: Higher direct images of dualizing sheaves. I
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cites work: Hyperbolic and Diophantine analysis
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cites work: A higher dimensional analogue of Mordell's conjecture over function fields
cites work: Hyperbolic fibre spaces and Mordell's conjecture over function fields
cites work: Q3658114
DOI: 10.2996/KMJ/1138038811
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description: scientific article; zbMATH DE number 4065162