  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.
