The existence of elliptic fibre space structures on Calabi-Yau threefolds
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.
scientific article; zbMATH DE number 734595
scientific article