Every compact Kähler threefold satisfying condition (C) admits a finite étale cover bimeromorphic to a fiber bundle over a torus fiber bundle, yielding a nowhere-vanishing holomorphic 1-form.
Kodaira dimension and zeros of holomorphic one-forms
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Holomorphic 1-forms without zeros on K\"ahler threefolds
Every compact Kähler threefold satisfying condition (C) admits a finite étale cover bimeromorphic to a fiber bundle over a torus fiber bundle, yielding a nowhere-vanishing holomorphic 1-form.