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.
Erratum and addendum to the paper: Abundance for K\"ahler threefolds
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abstract
In this text we signal a serious gap in the proof of the main theorem of our paper and explain which parts of the statement remain valid. In fact, the main theorem remains valid unless possibly the variety does not admit positive-dimensional subvarieties through a very general point and is not bimeromorphic to a quotient of a torus. This latter case would be ruled out by a Chern class inequality which holds in the algebraic case but is still unknown in the K\"ahler setting.
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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.