On compact analytic threefolds with non-trivial Albanese tori
type: Item
instance of: scholarly article
title: On compact analytic threefolds with non-trivial Albanese tori
zbMATH Open document ID: 0628.32037
DOI: 10.1007/BF01458060
published in: Mathematische Annalen
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164281
review text: Let \(f: V\to W\) be a fibre space (proper surjective morphism with connected fibres) of a n-dimensional compact connected manifold V over a m-dimensional compact connected manifold W and let denote by \(C_{n,m}\) the following conjecture: \(\kappa (V)\geq \kappa (W)+ \kappa (V_ x)\) where \(V_ x\) is a general fibre of f and \(\kappa\) ( ) means Kodaira dimension. The author proves conjecture \(C_{n,n-1}\) for fibre spaces of curves and conjecture \(C_{3,1}\) for fibre spaces of surfaces with positive Kodaira dimensions over a curve. For fibre spaces of surfaces with Kodaira dimension 0 the conjecture \(C_{3,1}\) is proved by using the theory of variation of Hodge structure except for the case when smooth fibres of f are isomorphic to complex tori of K3 surfaces and V is not in the class C (i.e. V is not bimeromorphically equivalent with a Kähler manifold). The results above are used to study the structure of the Albanese mappings of compact analytic threefolds with nonpositive Kodaira dimension. Let V be an analytic threefold and let \(\alpha\) : \(V\to A(V)\) be the Albanese mapping. Suppose \(\kappa (V)=0\). If \(\alpha\) is not surjective then \(\alpha\) (V) is a nonsingular curve of genus \(\geq 2\) and all the fibres of \(\alpha\) are connected. Smooth fibres are bimeromorphic to complex tori or K3 surfaces and almost all fibres of \(\alpha\) are nonalgebraic; V does not belong to \(class\quad C\). In case \(\alpha\) is surjective if \(\dim A(V)=3,\) then \(\alpha\) is bimeromorphic, if dim A(v)\(=2\) then \(\alpha\) is an elliptic threefold with constant moduli, if \(\dim A(V)=1\) then \(\alpha\) has connected fibres, a general fibre has Kodaira dimension 0 and if V belongs to the class C, then \(\alpha\) is bimeromorphic to an analytic fibre bundle. In the case \(\kappa (V)=- \infty\) let \(\beta\) : \(V\to W\), \(\gamma\) : \(W\to A(V)\) be the Stein factorization of \(\alpha\). Then \(\dim W\leq 2.\) If \(\dim W=2,\) the general fibres of \(\beta\) are \(…
reviewed by: Sorin Popescu
Mathematics Subject Classification ID: 32J15
Mathematics Subject Classification ID: 14J30
Mathematics Subject Classification ID: 32J25
zbMATH DE Number: 4021600
zbMATH Keywords: dimension 0
zbMATH Keywords: Albanese mappings of compact analytic threefolds
zbMATH Keywords: nonpositive Kodaira dimension
author: Kenji Ueno
MaRDI profile type: Publication
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OpenAlex ID: W1966596896
description: scientific article; zbMATH DE number 4021600