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

Birational positivity in dimension 4 (with an appendix by Fr\'ed\'eric Campana)