A Hermitian metric on a complex manifold is locally conformally Kähler (LCK) if it is conformal to a Kähler metric around each point.
Many of the complex non-Kähler manifolds admit such a metric. In this talk, I will discuss several examples, focusing on complex surfaces.
LCK manifolds provide a natural context for the study of twisted cohomology, also known as Morse-Novikov, which is the cohomology with values in a flat bundle. We compute it for Inoue surfaces and explain the relation between the twisted cohomology and their LCK geometry.
Links:
[1] http://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/node/3444
[3] http://www.mpim-bonn.mpg.de/node/3050