Numerical criteria for certain fiber spaces to be birationally trivial

Abstract

Let $f : X \to B$ be a fiber space over a curve $B$ whose general fiber $F$ belongs to one of the following type: 1) $F$ is of general type and satisfying some mild conditions, 2) $F$ is with trivial canonical sheaf. In this note, a numerical characterization for $f : X \to B$ to be birationally trivial is given.