These notes grew out of my graduate course at Hamburg University in the autumn of 2003. Their main purpose is to provide a quick and modern introduction to different aspects of Kähler geometry. I had tried to make the original lectures accessible to graduate students in mathematics and theoretical physics having only basic knowledge of calculus in several variables and linear algebra. The present notes should (hopefully) have retained this quality.
The text is organized as follows. The first part is devoted to a review of basic differential geometry. We discuss here topics related to smooth manifolds, tensors, Lie groups, principal bundles, vector bundles, connections, holonomy groups, Riemannian metrics, and Killing vector fields.
The reader familiar with the contents of a first course in differential geometry can pass directly to the second part, which starts with a description of complex manifolds and holomorphic vector bundles. Kähler manifolds are then discussed from the point of view of Riemannian geometry. This part ends with an outline of Hodge and Dolbeault theories, and a simple proof of the famous Kähler identities.
In the third part we study several aspects of compact Kähler manifolds: the Calabi conjecture, Weitzenböock techniques, Calabi–Yau manifolds, and divisors.
The material contained in each chapter is equivalent to a ninety-minute lecture. All chapters end with a series of exercises. Solving them may prove to be at least helpful, if not sufficient, for a reasonable understanding of the theory.
Acknowledgements. I would like to thank Christian Bär and the Department of Mathematics of Hamburg University for having invited me to teach this graduate course.