2012/08/31 by Massimiliano Pontecorvo, Pontecorvo, Massimiliano
Mathematics · #Algebraic Geometry and Number Theory #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #math.DG #msc:53C55
paper · pdf · doi:10.48550/arxiv.1208.6518
16 pages
arxiv created 2012/08/31 · arxiv updated 2012/09/03
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.