Global structure of an elliptic fibration.
type: Item
instance of: scholarly article
title: Global structure of an elliptic fibration.
zbMATH Open document ID: 1039.14004
author: Noboru Nakayama
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 2002-01-01T00:00:00Z
review text: In length and style, this paper is very much like a monograph. The author of this clearly structured and self-contained exposition has been an expert on elliptic fibrations for more than 15 years. Kodaira's theory of elliptic surfaces, which dates back to the 1960's, is a landmark in the study of elliptic fibrations. However, the higher-dimensional analytic case is more difficult and less clear. Recently, the local structure of projective elliptic fibrations was studied by \textit{N. Nakayama} [in: Higher dimensional birational geometry. Kyoto 1997, Adv. Stud. Pure Math. 35, 185--295 (2002; Zbl 1059.14015)]. The goal of the paper under review is a globalisation of these results. More specifically, the author aims at a description of the set of bimeromorphic equivalence classes of elliptic fibrations over a fixed base which are locally bimeromorphic to a projective morphism and which are equipped with a fixed marking. In addition, a reduced effective divisor in the base is fixed such that the open immersion of its complement is a toroidal embedding. All the fibrations are supposed to be bimeromorphically equivalent to a smooth fibration over this open complement. In the algebraic category, such equivalence classes are described by the Tate-Shafarevich group, which has an interpretation as an étale cohomology group. But, due to the presence of logarithmic transformations which produce multiple fibres and varieties which are not bimeromorphically equivalent, this is not sufficient in the complex analytic case. In order to treat multiple fibres, the author introduces the category of so-called \(\partial\)-spaces and equips it with a Grothendieck topology called the \(\partial\)-étale topology. If \(X\) is a normal complex variety and \(B\) a reduced Weil divisor on \(X\), it is interesting to observe that a reflexive sheaf on the \(\partial\)-space, which is determined by \((X,B)\), can be seen as a parabolic sheaf on \(X\). An important role in the study of elliptic fi…
reviewed by: Bernd Kreußler
Mathematics Subject Classification ID: 14D06
Mathematics Subject Classification ID: 14E30
Mathematics Subject Classification ID: 32J27
Mathematics Subject Classification ID: 14D07
Mathematics Subject Classification ID: 32G20
Mathematics Subject Classification ID: 14J27
Mathematics Subject Classification ID: 14J15
Mathematics Subject Classification ID: 14F20
zbMATH DE Number: 1833283
zbMATH Keywords: elliptic fibration
zbMATH Keywords: minimal model
zbMATH Keywords: variation of Hodge structure
zbMATH Keywords: toroidal embedding
zbMATH Keywords: Brauer group
zbMATH Keywords: Weierstrass model
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1145476270
OpenAlex ID: W2091283841
DOI: 10.2977/PRIMS/1145476270
description: scientific article; zbMATH DE number 1833283