Manifolds Containing an Ample P^1-bundle

TL;DR

This paper advances the classification of smooth projective varieties containing an ample P^d-bundle, confirming Sommese's conjecture under specific conditions and reducing it to a conjecture about projective space characterization.

Contribution

It proves Sommese's conjecture when the base variety has Picard rank one or is not uniruled, and links the conjecture to a broader characterization of projective space.

Findings

Confirmed the conjecture for Picard rank one bases.

Confirmed the conjecture for non-uniruled bases.

Reduced the conjecture to a characterization of projective space.

Abstract

Sommese has conjectured a classification of smooth projective varieties X containing, as an ample divisor, a P^d-bundle Y over a smooth variety Z. This conjecture is known if d>1, if dim(X)<5, or if Z admits a finite morphism to an Abelian variety. We confirm the conjecture if the Picard rank rho(Z)=1, 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, E is an ample vector bundle on W, and Hom(E, T_W) is non-zero, then W is isomorphic to P^n.