Complex and Kähler-manifolds

Here are some references that I have used in the past for various reasons. They are listed in no particular order.

• Huybrechts - Complex Geometry: An Introduction
• Moroianu - Lectures in Kähler Geometry (pdf version available here, but I believe the book has more details)
• Ballman - Lectures on Kähler Manifolds (pdf available here)
• Griffiths and Harris - Principles of Algebraic Geometry
• Demailly - Complex Algebraic and Analytic Geometry (pdf available here)
• Wells - Differential Analysis on Complex Manifolds

If I had to recommend a single book for you to consult for complex and Kähler geometry, I'd select Huybrechts' book. Having said that, complex and Kähler geometry are incredibly diverse areas, so it is hard to know exactly what it is you are looking for. I know that Huybrechts has at least a comment about SUSY and how it relates to the Kähler identities.

A one-dimensional complex manifold is called a Riemann surface. All Riemann surfaces are Kähler manifolds (you should try to learn why). For this reason, learning about Riemann surfaces is usually considered a good introduction to complex and Kähler geometry. Here are some books on Riemann surfaces, again, listed in no particular order.

• Forster - Lectures on Riemann Surfaces
• Donaldson - Riemann Surfaces (pdf available here, but I believe the book has more details)
• Ahlfors & Sario - Riemann Surfaces

Geometry, Topology, and Physics by Nakahara is a great reference with some good examples but it does not have a ton of Kahler material in it, still I have found it very helpful in my research. Something very complete is Lectures on Kahler Geometry by Moroianu. You can also find the Ballman lectures on Kahler geometry for free on the internet.