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Based on the celebrated result on zeros of holomorphic 1-forms on complexvarieties of general type by Popa and Schnell, we study holomorphic 1-forms on$n$-dimensional varieties of Kodaira dimension $n-1$. We show that a complexminimal smooth projective variety $X$ of Kodaira dimension $\kappa(X)=\dim X-1$admits a holomorphic 1-form without zero if and only if there is a smoothmorphism from $X$ to an elliptic curve. Furthermore, for a general smoothprojective variety (not necessarily minimal) $X$ of Kodaira codimension one, wegive a structure theorem for $X$ given that $X$ admits a holomorphic 1-formwithout zero.

Nowhere vanishing holomorphic one-forms on varieties of Kodaira codimension one