I was wondering if anyone knows any good references concerning Kähler manifolds and complex manifolds? I am studying supergravity theories and for the simplest $\mathcal{N}=1$ supergravity we will get these manifolds. Now course-notes are quite brief about these complex manifolds, so I was hoping someone on Physics SE might know a good (quite complete book) about the subject?
To get a rigorous mathematician's point of view, I've also posted this topic in on the math-stackexchange.
I strongly suggest Nakahara. Geometry, Topology and Physics.
There is a whole chapter in complex differential geometry and the Kahler case is treated well.
It is a good and clear introduction, written from a physicist and for physicists. However, it is not complete. With this I mean that if you want to have a strong knowledge of the subject (for example to work on it) you need some more than Nakahara.
But I'd give it a shot.
Chapter 0 of Griffiths and Harris, principles of algebraic geometry, gives a very good introduction in some 120 pages. In the remainder of the book the main focus is on complex algebraic varieties, which is a special, though still very broad, subclass.
I guess your needs are related to compactifications of supergravity theories. if this is true, then the book "Compact manifolds with special holonomy" by Joyce will be very useful. It has a section devoted to Kahler manifold since they indeed are of great importance for compactifications.
Then I'll suggest to look at review on flux compactifications, e.g. by M.Grana https://inspirehep.net/record/691224 . This describes geometry of manifolds with special geometry in application to physics (supergravity and phenomenology) while the book by Joyce contains more differential geometry.
Finally, recently I found this old paper https://inspirehep.net/record/16270 very useful. It has some discussion on Kahler manifolds as well.
You might find this excellent book entitled "Mirror Symmetry" by Hori et al, available online http://www2.maths.ox.ac.uk/cmi/library/monographs/cmim01.pdf, useful. Chapter 5, in particular, is a nice summary.
