### Birational positivity in dimension $4$

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.