A result of Popa and Schnell shows that any holomorphic 1-form on a smooth complex projective variety of general type admits zeros. More generally, given a variety $X$ which admits $g$ pointwise linearly independent holomorphic 1-forms, their result shows that $X$ has Kodaira dimension $\kappa(X) \leq \dim X - g$. In the extremal case where $\kappa(X) = \dim X - g$ and $X$ is minimal, we prove that $X$ admits a smooth morphism to an abelian variety, and classify all such $X$ by showing they arise as diagonal quotients of the product of an abelian variety with a variety of general type. The case $g = 1$ was first proved by the third author, and classification results about surfaces and threefolds carrying nowhere vanishing forms have appeared in work of Schreieder and subsequent joint work with the third author. We also prove a birational version of this classification which holds without the minimal assumption, and establish additional cases of a conjecture of the third author.