  We assume that the existence and termination conjecture for flips holds. A
complex projective manifold is said to be {\it of almost general type} if the
intersection number of the canonical divisor with every very general curve is
strictly positive. Let $f$ be an algebraic fiber space from $X$ to $Y$. Then
the manifold $X$ is of almost general type if every very general fiber $F$ and
the base space $Y$ of $f$ are of almost general type.
