On a conjecture of Beltrametti and Sommese

Abstract

Let X X be a projective manifold of dimension n n . Beltrametti and Sommese conjectured that if A A is an ample divisor such that K X + ( n − 1 ) A K_X+ (n-1)A is nef, then K X + ( n − 1 ) A K_X+(n-1)A has non-zero global sections. We prove a weak version of this conjecture in arbitrary dimension. In dimension three, we prove the stronger non-vanishing conjecture of Ambro, Ionescu and Kawamata and give an application to Seshadri constants.