On a question of Vaisman concerning complex surfaces

Abstract

The last years have seen striking improvements on Vaisman's question about existence of locally conformally K\"ahler (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 K\"ahler geometry in four-dimension and prove a new result concerning generalized K\"ahler structures on Hyperbolic Inoue surfaces. We conclude with a simple observation on a question of Brunella.