  We prove that the abundance conjecture holds on a variety $X$ with mild
singularities if $X$ has many reflexive differential forms with coefficients in
pluricanonical bundles, assuming the Minimal Model Program in lower dimensions.
This implies, for instance, that under this condition, hermitian semipositive
canonical divisors are almost always semiample, and that klt pairs whose
underlying variety is uniruled have good models in many circumstances. When the
numerical dimension of $K_X$ is $1$, our results hold unconditionally in every
dimension. We also treat a related problem on the semiampleness of nef line
bundles on Calabi-Yau varieties.
