  In this article we study how the birational geometry of a normal projective
variety $X$ is influenced by a normal subvariety $A \subset X.$ One of the most
basic examples in this context is provided by the following situation. Let
$f:X\to Y$ be a surjective holomorphic map with connected fibers between
compact connected complex manifolds. It is well known that given a general
fiber $A$ of $f$ we have $$ \kappa(X)\le \kappa(A)+\dim Y. $$ This article grew
out of the realization that this result should be true with $\dim Y$ replaced
by the codimension $\cod_X A$ for a pair $(X,A)$ consisting of a normal
subvariety $A$ of a compact normal variety $X$ under weak semipositivity
conditions on the normal sheaf of $A$ and the weak singularity condition
$\cod_A (A\cap\sing X)\ge 2$. We shall now state our main results in the
special case of a submanifold $A$ in a projective manifold $X$ and we also
simplify the semipositivity notion.
