An Introduction to Complex, Kähler and Locally Conformal Kähler Geometry

In this note we briefly summarize the necessary tools from complex manifold theory in order to give an introduction to the basic results about Kähler and locally conformal Kähler manifolds. We look at many equivalent definitions of all three types of manifolds and examine in detail the parallels which arise in the theories. These parallels include the compatibility of metrics and fundamental 2-form under exterior differentiation and the equivalence of the Levi-Civita and the Chern connection in the complex Kähler case. Some examples are provided to motivate the subjects and to demonstrate that these classes are all properly nested within each other. With a foundation of complex and Kähler manifolds, we further sort these classes by introducing the notion of a manifold being locally conformal Kähler. We discuss existence and uniqueness of specific linear connections on these manifolds and provide formulae expressing them in terms of known objects arising from the metric.