On a question of Vaisman concerning complex surfaces

TL;DR

This paper reviews recent progress on Vaisman's question about lcK metrics on compact complex surfaces, explores connections with generalized K"ahler geometry, and presents new results on Hyperbolic Inoue surfaces.

Contribution

It provides a comprehensive review of known results on lcK metrics, establishes a new link with generalized K"ahler structures, and proves novel results for Hyperbolic Inoue surfaces.

Findings

Complete classification of lcK metrics on known compact complex surfaces

Established a relation between lcK surfaces and generalized K"ahler geometry in four dimensions

Proved new results on generalized K"ahler structures on Hyperbolic Inoue 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.