On manifolds with trivial logarithmic tangent bundle
type: Item
instance of: scholarly article
title: On manifolds with trivial logarithmic tangent bundle
zbMATH Open document ID: 1058.32011
author: Jörg Winkelmann
published in: Osaka Journal of Mathematics
publication date: 2004-10-28T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.ojm/1153493523
review text: The automorphism group \(\text{Aut}(X)\) of a compact complex manifold \(X\) is a finite dimensional complex Lie group with Lie algebra the holomorphic vector fields on \(X\). Since the vector fields are tangent to the Aut\((X)\)-orbits, a connected compact complex manifold \(X\) with trivial holomorphic tangent bundle \(T(X)\) is a quotient of \(\text{Aut}(X)\) by a uniform discrete subgroup. If \(X\) is Kähler, then \(X=\text{Aut}^0(X)\) is a compact complex torus [see \textit{H. C. Wang}, Proc. Am. Math. Soc. 5, 771--776 (1954; Zbl 0056.15403)]. The author considers the situation of a non-compact complex manifold \(U\) with holomorphically trivial tangent bundle \(T(U)\) which is Zariski-open in a compact complex manifold \(X\) of class \({\mathcal C}\), i.e. \(X\) is bimeromorphic to a compact Kähler manifold. If \(D:=X\backslash U\) is a divisor with only simple normal crossings the analytic sheaf \(\Omega^1(\log D)\) of logarithmic 1-forms with respect to \(D\) on \(X\) is locally free. Its dual bundle, the logarithmic tangent bundle \(T(-\log D)\), is an extension of \(T(U)\) to \(X\) [cf. \textit{D. Mumford}, Invent. Math. 42, 239--272 (1977; Zbl 0365.14012)]. The main result of the paper under review is the equivalence of the conditions (1) ``\(D\) is a divisor with only simple normal crossings and \(T(-\log D)\) is holomorphically trivial'', and (2) ``\(\text{Aut}(X)\) contains a complex semi-torus \(G\) such that \(G\) operates freely and transitively on \(U\) and all isotropy subgroups of \(G\) in \(D\) are complex semi-tori as well''. Here a complex semi-torus is defined to be a quotient of \((\mathbb C^n,+)\) by a discrete subgroup \(\Gamma\) with \(\mathbb C\Gamma=\mathbb C^n\). The class \({\mathcal C}\) condition is needed in the proof \((1)\Rightarrow (2)\) to guarantee that the logarithmic 1-forms on \(X\) are closed. There are many examples of toric varieties with an equivariant compactification by a divisor \(D\) such that (1) is true, especiall…
reviewed by: Eberhard Oeljeklaus
Mathematics Subject Classification ID: 32J27
Mathematics Subject Classification ID: 32M12
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 14M25
zbMATH DE Number: 2111467
zbMATH Keywords: logarithmic tangent bundle
zbMATH Keywords: logarithmic 1-form
zbMATH Keywords: complex semi-torus
MaRDI profile type: Publication
Recommended article: Automorphism groups of $\mathbb{P}^1$-bundles over a non-uniruled base
Recommended article: A characterization of compact complex tori via automorphism groups
Recommended article: On manifolds with trivial logarithmic tangent bundle: the non-Kähler case
Recommended article: Dolbeault cohomology of compact complex homogeneous manifolds
Recommended article: Holomorphic splittings associated with holomorphic complex torus actions
Recommended article: Q4552342
Recommended article: Bimeromorphic automorphism groups of certain \(\mathbb{P}^1\)-bundles
Recommended article: Homogeneous principal bundles over manifolds with trivial logarithmic tangent bundle
Recommended article: Semisimplicity of the group of biholomorphisms of the universal covering of a compact complex manifold with ample canonical bundle
Recommended article: Invariant Hodge forms and equivariant splittings of algebraic manifolds
description: scientific article; zbMATH DE number 2111467