Belgun, Florin Alexandru On the metric structure of non-Kähler complex surfaces. (English) Zbl 0988.32017 Math. Ann. 317, No. 1, 1-40 (2000). 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. Reviewer: Guy Roos (Paris-St.Peterburg) Cited in 1 ReviewCited in 109 Documents 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.) Keywords:compact complex surfaces; non-Kähler complex surfaces PDFBibTeX XMLCite \textit{F. A. Belgun}, Math. Ann. 317, No. 1, 1--40 (2000; Zbl 0988.32017) Full Text: DOI