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Elliptic surfaces and ample vector bundles

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 pg(Z) > 0 unless X is a Pn−2-bundle over a smooth surface S with pg(S) = 0.