Vaisman & Locally Conformally Kähler Geometry

The Lee group of a Vaisman manifold is the complex Lie group generated by the Lee vector field. It depends only on the complex structure of the manifold, and has properties which uniquely identify it as a subgroup of the biholomorphism group. This allows one for instance to show that among all Hopf manifolds, only the diagonal ones admit Vaisman metrics.  

I will survey different properties related to the Lee group, particularly focusing on how to interpret its compactness. Then I will explain how special deformations of the complex structure produce a compact Lee group. Finally, I will point out different conditions which imply a priori compactness of the Lee group. 

I shall review two recent results obtained jointly with Misha Verbitsky concerning the existence of the minimal model for comapct LCK manifolds with potential and the non-existence of balanced metrics on all compact complex manifolds bimeromorphic with known LCK manifolds.

We discuss cohomology properties of the known classes of examples of locally conformally Kähler manifolds.

According to Fujiki and Donaldson's foundational work, the scalar curvature of Kähler metrics arises as a moment map for an infinite-dimensional Hamiltonian action. In this talk, we generalize this result to the broader framework of locally conformally Kähler Geometry. This is joint work with D. Angella, S. Calamai, and C. Spotti.

Inspired by a classical results of Tian on approximations of Kähler metrics on compact manifolds by Kodaira embeddings, we will discuss the analogous problem for compact Vaisman manifolds into their model spaces, that is, Hopf manifolds. We will relate this result to several embedding results for compact Vaisman manifolds and their relation to Sasakian immersions.