Closed holomorphic 1-forms without zeros on Stein manifolds (Q2458874)

The author proves that if \(X\) is a Stein manifold, every cohomology class in the cohomology group \(H^{1} (X, \mathbb{C})\) is represented by a closed holomorphic 1-form (this was already known) which is without zeros (and this is the novelty in this paper). This result is not true for arbitrary complex manifolds, and the author furnishes a counterexample. The author proves her result by starting with a closed 1-form \(\theta \) on \(X\) and by constructing a closed 1-form \(\widetilde {\omega}\) without zeros and which has the same cohomology class as \(\theta\) in \(H^{1} (X, \mathbb{C})\). If \(K\) is a compact subset of \(X\), as a closed nowhere vanishing holomorphic 1-form on \(X\) defines a holomorphic foliation \(\mathcal F\) of codimension 1, the author furnishes an approximation theorem for holomorphic nonsingular hypersurface foliations, which is done by a global foliation \(\widetilde{\mathcal F}\) on \(X\) such that \(\widetilde{\mathcal F}\) and \(\mathcal F\) are conjugate to each other on a neighborhood of \(K\).