Elliptic surfaces and ample vector bundles

Antonio Lanteri and Hidetoshi Maeda

Vol. 200 (2001), No. 1, 147–157

DOI: 10.2140/pjm.2001.200.147

Abstract

Let ℰ be an ample vector bundle of rank n− 2 ≥ 2 on a complex projective manifold X of dimension n having a section whose zero locus is a smooth surface Z. We determine the structure of pairs (X,ℰ) as above under the assumption that Z is a properly elliptic surface. This generalizes known results on threefolds containing an elliptic surface as a smooth ample divisor. Among the applications we prove a conjecture relating the Kodaira dimension of X to that of Z, and we show that if 0 ≤ κ(Z) ≤ 1, then p_g(Z) > 0 unless X is a ℙⁿ⁻²-bundle over a smooth surface S with p_g(S) = 0.

Milestones

Received: 21 October 1999

Published: 1 September 2001