Manifolds with nef anticanonical bundle

Abstract

Abstract Let X be a compact Kähler manifold such that the anticanonical bundle - K X $-K_{X}$ is nef. A classical conjecture claims that the Albanese map X → T $X\to T$ is submersive. We prove this conjecture if the general fibre is a weak Fano manifold. If X is projective we prove the conjecture also for fibres of dimension at most two.