# Complex and Kähler-manifolds

I was woundering if anyone knows any good references about Kähler and complex manifolds? I'm studying supergravity theories and for the simpelest N=1 supergravity we'll get these. Now in the course-notes the're quite short about these complex manifolds. I was hoping someone of you guys might know a good (quite complete book) about the subject ?

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I've also posted this on the physics stackexchange-site, hoping that maybe a string-theorist of supergravity-specialist might be able to provide some information. But from what I'm seeing in the posts here the answers are already very nice! A big thanks in advance, I'll be going trough the sources somwhere in the end of this week or the beginning of next week somewhere!

• Grifith and Harris. – Gil Bor Jan 8 '14 at 0:34

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
• To add to the Riemann surfaces list, if the OP has access to a library, they should look at Gunning's Lectures on Riemann Surfaces. – user98602 Jan 8 '14 at 2:00
• @Michael Albanese, looks like a very nice list! I'll go trough it as soon as possible! Do you have any hints wich one to go trough first? My background: I've got the "sloppy physicist's first encounter to differential geometry". – Nick Jan 8 '14 at 11:47
• I would recommend having a look at Lee's Introduction to Smooth Manifolds if you want to improve your differential geometry skills. Huybrechts' book is likely to cover whatever you need to know. – Michael Albanese Jan 8 '14 at 11:49
• @MichaelAlbanese thanks a lot! My biggest problem is (I guess) that I can apply the formulas of differential geometry correctly, but for example I don't understand quite how we can relate partial derivatives with the vectors we know from linear algebra. Somehow I know it makes sence to look at the functions on a manifold to characterise it, but not how it relates to vectors (something tells me the unit-vectors are given by the coördinate curves). – Nick Jan 8 '14 at 15:11
• @Dominique "(something tells me the unit-vectors are given by the coördinate curves" Usually not. If you have a Riemannian manifold and a coordinate system that gives unit tangent vectors in a neighborhood of a point, the curvature is flat in that neighborhood. – Gunnar Þór Magnússon Jun 2 '16 at 18:00

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.