The existence of elliptic fibre space structures on Calabi-Yau threefolds
type: Item
instance of: scholarly article
title: The existence of elliptic fibre space structures on Calabi-Yau threefolds
zbMATH Open document ID: 0819.14015
DOI: 10.1007/BF01450510
author: P. M. H. Wilson
published in: Mathematische Annalen
publication date: 1995-08-31T00:00:00Z
full work available at URL: https://eudml.org/doc/165276
review text: The existence of an elliptic fibre space structure, on a Calabi-Yau threefold \(X\), implies -- in particular -- the existence of a nef integral divisor \(D\) on \(X\) with \(D^ 3 = 0\), \(D^ 2 \not \equiv 0\) and \(D.c_ 2 \geq 0\). The question is to prove, or to reject, the converse: Does the existence of a nef integral \(D\), on the Calabi-Yau \(X\), such that \(D^ 3 = 0\), \(D^ 2 \not \equiv 0\) and \(D.c_ 2 \geq 0\), imply the existence of an elliptic fibre structure on \(X\)? As it follows from previous results of the author [Invent. Math. 98, No. 1, 139-155 (1989; Zbl 0688.14032)], \textit{K. Oguiso} [``On certain rigid fibered Calabi-Yau threefolds'' (Preprint, MPI Bonn 1993)], \textit{N. I. Shepherd-Barron} and the author [J. Algebr. Geom. 3, No. 2, 265-281 (1994; Zbl 0807.14031)], and \textit{T. Peternell} [Math. Z. 207, No. 2, 305-318 (1991; Zbl 0735.14028)], the answer is always Yes -- if \(D.c_ 2 > 0\). In this paper the author gives a positive answer of this question -- also in the case when \(D.c_ 2 = 0\), provided \(X\) fulfills some additional topological restrictions.
Mathematics Subject Classification ID: 14J30
Mathematics Subject Classification ID: 14J15
Mathematics Subject Classification ID: 14J10
zbMATH DE Number: 734595
zbMATH Keywords: elliptic fibre space
zbMATH Keywords: Calabi-Yau threefold
zbMATH Keywords: nef integral divisor
reviewed by: Atanas Iliev
MaRDI profile type: Publication
cites work: Q4023663
cites work: Divisorenklassengruppen quasihomogener Singularitäten.
cites work: The cone of curves of algebraic varieties
cites work: Crepant blowing-up of 3-dimensional canonical singularities and its application to degenerations of surfaces
cites work: Flops
cites work: Q4012871
cites work: Calabi-Yau manifolds and a conjecture of Kobayashi
cites work: ON ALGEBRAIC FIBER SPACE STRUCTURES ON A CALABI-YAU 3-FOLD
cites work: On certain rigid fibered Calabi-Yau threefolds
cites work: Q4294357
cites work: Calabi-Yau manifolds with large Picard number
cites work: The Kähler cone on Calabi-Yau threefolds
description: scientific article; zbMATH DE number 734595