Principles of Locally Conformally Kahler Geometry

An LCK (locally conformally Kahler) manifold is a complex manifold admitting a Kahler covering with monodromy acting by homotheties. Hopf manifolds and their submanifolds are the prime examples. This book presents an introduction to the principles of LCK geometry (the first two parts) and its current situation (the last part). It is supposed to be accessible to master and graduate students of complex geometry. The book contains many exercises of different levels of difficulty. We finish it by a list of open questions.Comment: 776 pages, 14 figures, version 4.0, Latex. Many small changes introduced, additions to bibliography and exercises. Chapter 3 rewritte

Kobayashi pseudometric on hyperkähler manifold and Kobayashi’s conjecture

The Kobayashi pseudometric on a complex manifold M is the maximal pseudometric such that any holomorphic map from the Poincare disk to M is distance-decreasing. Kobayashi has conjectured that this pseudometric vanishes on Calabi-Yau manifolds. Using ergodicity of complex structures, we prove this conjecture for any hyperkähler manifold that admits a deformation with a Lagrangian fibration, and its Picard rank is not maximal. We shall discuss the proof of Kobayashi’s conjecture for K3 surfaces and for certain hyperkähler manifolds. These results are joint with S. Lu and M. Verbitsky.Non UBCUnreviewedAuthor affiliation: Stony Brook UniversityPostdoctora