Kodaira dimension of subvarieties II

This paper continues the study of non-general type subvarieties begun in a joint paper with M.Schneider and A.Sommese (Int.L.Math. 10, 1999). We prove uniruledness of a projective manifold containing a submanifold not of general type whose normal bundle has positivity properties and moreover study the rational quotient. We also relate the fundamental groups and prove a cohomological criterion for a manifold to be rationally connected.