On semiample vector bundles and parallelizable compact complex manifolds

We provide a characterization of parallelizable compact complex manifolds and their quotients using holomorphic symmetric differentials. In particular we show that compact complex manifolds of Kodaira dimension 0 having strongly semiample cotangent bundle are parallelizable manifolds, while compact complex manifolds of Kodaira dimension 0 having weakly semiample cotangent bundle are quotients of parallelizable manifolds. The main constructions used involve considerations about semiampleness of vector bundles, which are themselves of interest. As a byproduct we prove that compact manifolds having Kodaira dimension 0 and weakly sermiample cotangent bundle have infinite fundamental group, and we conjecture that this should be the case for all compact complex manifolds not of general type with weakly semiample cotangent bundle.