# Kähler and complex manifolds

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.

• Maybe these lectures (Chapter 4). – Trimok Jan 8 '14 at 12:11
• Would Mathematics be a better home for this question? – Emilio Pisanty Jan 8 '14 at 13:18
• @EmilioPisanty I also have a copy of this question in the mathematics-part of the forum (math.stackexchange.com/q/630838). But I figured that maybe a physicist point of view might also be helpful ? – Nick Jan 8 '14 at 15:05
• In that case, you should always indicate the fact that you've cross-posted, in both posts. – Emilio Pisanty Jan 8 '14 at 16:00
• @EmilioPisanty Edited it! :) – Nick Jan 8 '14 at 17:16

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.

• 120 pages, seems like quite a long way to go. Are you perhaps familiar with some works that might give a quicker acces (given that I have had e "first encounter" with differential geometry ?) – Nick Jan 8 '14 at 11:45
• I think it could still be useful. It is broken up in 7 sections that can probably be skipped entirely when they treat something you know already. Kaehler manifolds are only introduced in the last section, which is just over 20 pages. If you already know complex geometry including sheaf cohomology you can start there, otherwise you'll have to go through more of them. With your differential geometry background you can probably go very fast through at least two of the other 6. – doetoe Jan 10 '14 at 23:35

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.