Birational positivity in dimension 4 (with an appendix by Frédéric Campana)

Abstract

In this paper we prove that for a nonsingular projective variety of dimension at most 4 and with non-negative Kodaira dimension, the Kodaira dimension of coherent subsheaves of ${\Omega }^p$ is bounded from above by the Kodaira dimension of the variety. This implies the finiteness of the fundamental group for such an $X$ provided that $X$ has vanishing Kodaira dimension and non-trivial holomorphic Euler characteristic.

To be published in Annales de l'institut Fourier, Vol 63 (2013)