Manifolds containing an ample P1-bundle

Sommese has conjectured a classification of smooth projective varieties X containing, as an ample divisor, a -bundle Y over a smooth variety Z. This conjecture is known if , if , or if Z admits a finite morphism to an Abelian variety. We confirm the conjecture if the Picard rank , or if Z is not uniruled. In general we reduce the conjecture to a conjectural characterization of projective space: namely that if W is a smooth projective variety, is an ample vector bundle on W similar or equal to P-n.