Principles of Locally Conformally Kähler Geometry

Introduction.- Part I: Lectures in locally conformally Kähler geometry.- Kähler manifolds.- Connections in vector bundles and the Froebenius theorem.- Locally conformally Kahler manifolds.- Hodge theory on complex manifolds and Vaisman's theorem.- Holomorphic vector bundles.- CR, Contact and Sasakian manifolds.- Vaisman manifolds.- The structure of compact Vaisman manifolds.- Orbifolds.- Quasi-regular foliations.- Regular and quasi-regular Vaisman manifolds.- LCK manifolds with potential.- Embedding LCK manifolds with potential in Hopf manifolds.- Logarithms and algebraic cones.- Pseudoconvex shells and LCK metrics on Hopf manifolds.- Embedding theorem for Vaisman manifolds.- Non-linear Hopf manifolds.- Morse-Novikov and Bott-Chern cohomology of LCK manifolds.- Existence of positive potentials.- Holomorphic S^1 actions on LCK manifolds.- Sasakian submanifolds in algebraic cones.- Oeljeklaus-Toma manifolds.- Appendices.- Part II: Advanced LCK geometry.- Non-Kähler elliptic surfaces.- Kodaira classification for non-Kähler complex surfaces.- Cohomology of holomorphic bundles on Hopf manifolds.- Mall bundles and flat connections on Hopf manifolds.- Kuranishi and Teichmüller spaces for LCK manifolds with potential.- The set of Lee classes on LCK manifolds with potential.- Harmonic forms on Sasakian and Vaisman manifolds.- Dolbeault cohomology of LCK manifolds with potential.- Calabi-Yau theorem for Vaisman manifolds.- Holomorphic tensor fields on LCK manifolds with potential.- Part III: Topics in locally conformally Kähler geometry.- Twisted Hamiltonian actions and LCK reduction.- Elliptic curves on Vaisman manifolds.- Submersions and bimeromorphic maps of LCK manifolds.- Bott-Chern cohomology of LCK manifolds with potential.- Hopf surfaces in LCK manifolds with potential.- Riemannian geometry of LCK manifolds.- Einstein-Weyl manifolds and the Futaki invariant.- LCK structures on homogeneous manifolds.- LCK structures on nilmanifolds and solvmanifolds.- Explicit LCK metrics on Inoue surfaces.- More on Oeljeklaus-Toma manifolds.- Locally conformally parallel and non-parallel structures.- Open questions.