Manifolds containing an ample $$\mathbb {P}^1$$-bundle

Abstract

Sommese has conjectured a classification of smooth projective varieties X containing, as an ample divisor, a \(\mathbb {P}^d\)-bundle Y over a smooth variety Z. This conjecture is known if \(d>1\), if \(\dim (X)\le 4\), 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, \(\mathscr {E}\) is an ample vector bundle on W, and \(\text {Hom}(\mathscr {E}, T_W)\not =0\), then \(W\simeq \mathbb {P}^n\).