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Locally conformal Hermitian metrics on complex non-Kähler manifolds. (English) Zbl 1353.32026

In this interesting paper, the authors investigate complex non-Kähler manifolds with compatible metrics being locally conformal to metrics with special cohomological properties. The study is focussed on locally conformal holomorphic-tamed structures. The authors provide results and examples for which the existence of these structures assures the existence of a locally conformal Kähler structure. This happens for example for 6-dimensional nilmanifolds endowed with a left-invariant complex structure, while the Inoue surface \(S^{\pm}\) provides a counterexample.

MSC:

32Q99 Complex manifolds
32Q15 Kähler manifolds
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