Algebraic fiber space whose generic fiber and base space are of almost general type

Abstract

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.