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On the metric structure of non-Kähler complex surfaces. (English) Zbl 0988.32017

Let \(X\) be a compact complex surface, i.e. a compact \(2\)-dimensional complex manifold. For a Hermitian metric \(h\) on \(X\) and its associated Kähler form \(\omega\), the Lee form \(\theta\) is defined by \(d\omega=-2\theta\wedge\omega\).
A compact complex surface admits a Kähler metric if and only if its first Betti number \(b_{1}(X)\) is even. When \(b_{1}(X)\) is odd, the author gives, through case-by-case verification on the Kodaira classification of compact complex surfaces:
1) the classification of non-Kähler surfaces which are locally conformally Kähler, or equivalently admit a Hermitian metric such that the Lee form \(\theta\) is closed;
2) the classification of locally conformally Kähler surfaces such that the Lee form \(\theta\) can be made parallel with respect to \(h\) by a conformal change of the metric.
The author also classifies complex surfaces admitting locally homogeneous locally conformally Kähler structure.

MSC:

32J27 Compact Kähler manifolds: generalizations, classification
32Q57 Classification theorems for complex manifolds
53C55 Global differential geometry of Hermitian and Kählerian manifolds
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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