Logarithmic transformations on elliptic fiber spaces. II: Elliptic bundle case
type: Item
instance of: scholarly article
title: Logarithmic transformations on elliptic fiber spaces. II: Elliptic bundle case
zbMATH Open document ID: 0692.14009
DOI: 10.1007/BF01444550
author: Yoshio Fujimoto
published in: Mathematische Annalen
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164752
review text: [For part I see J. Math. Kyoto Univ. 28, No.1, 91-110 (1988; Zbl 0654.14021).] Kodaira introduced the notion of logarithmic transformations and showed that a non-Kähler elliptic surface and a Kähler one can be transformed to each other via logarithmic transformations. The main purpose of this paper is to prove the counterpart of Kodaira's theorem for elliptic bundles. Here by an elliptic bundle f: \(Y\to X\) over X, we mean that Y is a principal fiber bundle over a complex manifold X whose typical fiber and structure groups are non-singular elliptic curves. For an elliptic bundle f: \(Y\to X\) over X, we shall define generalized logarithmic transformations along an arbitrary \(Cartier\quad divisor\quad D\) on X, which is not necessarily assumed to be reduced, irreducible or effective. We state our main theorem: Theorem (A). Let f: \(Y\to X\) be an elliptic bundle over a projective manifold X, which satisfies either of the following conditions: (1) Y is Kähler; (2) \(h^{2,0}(X):=\dim (H^ 0(X,\Omega^ 2_ X))=0.\) Then Y can be obtained from the trivial elliptic bundle over X by a succession of generalized logarithmic transformations. However, theorem (A) does not necessarily hold if we drop the assumption that Y is Kähler. We can give the characterization of elliptic bundles which are obtained from the trivial bundle by generalized logarithmic transformations.
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 14J27
Mathematics Subject Classification ID: 14E05
zbMATH DE Number: 4132324
zbMATH Keywords: logarithmic transformations
zbMATH Keywords: elliptic bundle
MaRDI profile type: Publication
cites work: A class of compact, complex manifolds which are not algebraic
cites work: On automorphism groups of compact Kähler manifolds
cites work: Logarithmic transformations on elliptic fiber spaces
cites work: Pluricanonical mappings of elliptic fiber spaces
cites work: An intrinsic characterization of Kähler manifolds
cites work: Q3715427
cites work: On elliptic surfaces in characteristic p
cites work: On the Structure of Compact Complex Analytic Surfaces, I
cites work: On compact analytic surfaces. II
cites work: Kähler metrics on elliptic surfaces
cites work: Q4744383
cites work: On compact analytic threefolds with non-trivial Albanese tori
cites work: Stabilité de la classe des variétés Kaehleriennes par certains morphismes propres
Recommended article: Logarithmic transformations on elliptic fiber spaces
Recommended article: Q3362154
Recommended article: LOGARITHMIC TRANSFORMATION OF AN ELLIPTIC SURFACE, AND VECTOR BUNDLES
Recommended article: Q4280966
Recommended article: Logarithmic transformations of rigid analytic elliptic surfaces
Recommended article: On transformations of elliptic spaces
Recommended article: A note on logarithmic transformations on the Hopf surface
Recommended article: \(C^{\infty}\)-logarithmic transformations and generalized complex structures
Recommended article: Fourier-Mukai transforms and stable bundles on elliptic curves
Recommended article: Logarithmic connections on principal bundles over a Riemann surface
description: scientific article; zbMATH DE number 4132324