Subvarieties of parallelizable manifolds
type: Item
instance of: scholarly article
title: Subvarieties of parallelizable manifolds
zbMATH Open document ID: 0824.32011
DOI: 10.1007/BF01444897
author: Jörg Winkelmann
author: Alan Huckleberry
published in: Mathematische Annalen
publication date: 1994-03-23T00:00:00Z
full work available at URL: https://eudml.org/doc/165052
review text: A compact complex manifold \(X\) is called parallelizable if the complex tangent bundle \(TX\) is holomorphically trivial. Equivalently, the complex Lie group of automorphisms \(G\) acts trivially and locally freely on \(X\), so that \(X \simeq G/ \Gamma\), where \(\Gamma\) is a discrete subgroup. The goal of the paper under review is to study subvarieties \(Z\) of complex parallelizable manifolds. The case of codimension one subvarieties being easy (and known), the authors study subvarieties of higher codimension. They consider a whole spectrum of related problems, such as existence of subvarieties, existence of nontrivial holomorphic vector bundles, number of closed orbits, etc. However their main interest is in studying the pluricanonical map of \(Z\) from a group theoretic viewpoint. An interesting application is that if \(G\) is simple, then \(\text{codim} Z \geq \sqrt {\text{dim}}G\), that is, the dimension of \(Z\) is bounded from above. Finally, the authors prove a version of Bloch's conjecture: if \(f : \mathbb{C} \to X\) is a holomorphic map, then \(\overline {f(\mathbb{C})}\) is an orbit of a complex Lie subgroup of \(G\).
Mathematics Subject Classification ID: 32M05
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 32M10
Mathematics Subject Classification ID: 32L05
zbMATH DE Number: 537219
zbMATH Keywords: homogeneous space
zbMATH Keywords: automorphisms
zbMATH Keywords: subvarieties
zbMATH Keywords: complex parallelizable manifolds
zbMATH Keywords: higher codimension
zbMATH Keywords: Bloch's conjecture
zbMATH Keywords: orbit of a complex Lie subgroup
reviewed by: F. L. Zak
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 537219