On Hermitian geometry of complex surfaces. (English) Zbl 1085.53065

Kowalski, Oldřich (ed.) et al., Complex, contact and symmetric manifolds. In honor of L. Vanhecke. Selected lectures from the international conference “Curvature in Geometry” held in Lecce, Italy, June 11–14, 2003. Boston, MA: Birkhäuser (ISBN 0-8176-3850-4/hbk). Progress in Mathematics 234, 153-163 (2005).
The authors give a very nice exposition of locally conformal Kähler metrics on complex surfaces. A Hermitian surface has a natural non-degenerate \((1,1)\)-form \(\omega\). One can conclude that there is a \(1\)-form \(\theta\) such that \(d\omega=\omega\wedge\theta\). This form is called the Lee form. The manifold is locally conformal Kähler exactly when this form is closed.
After some introduction the authors carefully go through the Enriques-Kodaira classification of surfaces quoting results of others showing that the existence of locally conformal Kähler metrics is completely understood except for surfaces of class \(VII_0\). In the last section of this paper, the authors describe their results. The final one stating that every minimal hyperbolic of half Inoue surface with \(b_2=m\) admits an \(m\)-dimensional family of anti-self-dual Hermitian metrics. Together with a result for some Enoki surfaces. The proof of this result will appear elsewhere, but the outline indicates that the authors use twistor methods. This is a nice introduction to an area with some interesting open problems.
For the entire collection see [Zbl 1062.53001].


53C55 Global differential geometry of Hermitian and Kählerian manifolds