On a question of Vaisman concerning complex surfaces
arXiv:1208.6518
Abstract
The last years have seen striking improvements on Vaisman's question about existence of locally conformally Kähler (lcK) metrics on compact complex surfaces. The aim of this paper is two-fold. We review results of different authors which, for all known examples of compact complex surfaces, give a complete answer to Vaisman's question. We also point out a relation between lcK surfaces and generalized Kähler geometry in four-dimension and prove a new result concerning generalized Kähler structures on Hyperbolic Inoue surfaces. We conclude with a simple observation on a question of Brunella.
16 pages