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I am looking for references on two distinct (though related) topics.

  1. Potential theory :

    I read some time ago the book of Ransford (Potential Theory in the complex plane). It was great (intuitive and relatively elementary) but now insufficient for my purposes. In particular I'd like a book that works on a more general setting (manifolds, more than 1 variable). Anyone has a good one ?

  2. Complex geometry

    I realize that is a vast topic. I already have a background in differential geometry, a bit of Riemann surfaces. What I need to learn about is stuff like : $\overline{\partial}$ and $dd^c$ operators, quadratic differentials on a complex manifold, residues of complex forms... I know this is not very precise but hey, if I knew exactly what I were looking for I wouldn't need to ask...

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2 Answers 2

up vote 4 down vote accepted

I note that you wrote complex differential geometry. There is a difference between complex differential and analytic geometry, the latter having much in common with algebro-geometric scheme theory. However, there is no schism between the two - if you're interested in one, you'd better learn a hell of a lot about the other.

For the former, I feel comfortable recommending:

  1. Zheng - Complex differential geometry : A very diffeo-geometrical introduction to the subject. Does not go into extreme technical details, but does not shy away from difficulties.

  2. Demailly - Complex analytic and differential geometry (available for free on Demailly's website) : This is where you'll find all the technical details. Amazing for a second run on the subject, trying but ultimately rewarding on the first. (Note: Demailly recommends Hörmander's book for the complex analytic technical details needed for his own.)

There are books by Werner Ballmann and Andrei Moroianu on Kähler geometry as well. Both are good. Claire Voisin's book on Hodge theory is closely related to what you want, but more algebraic. I imagine you'll also want to look through Shafarevich's books on algebraic geometry, Mumford's books on the same, and anything written by Joe Harris for motivation and examples. Then you'll want to look at Kobayashi-Nomizu as well.

The problem, as always, is that there are seven million different important things to learn. Thus it is pretty much impossible to write a textbook that trails a coherent narrative and covers all these subjects. Also, beware that a "second course" type book on complex differential geometry does, to my knowledge, not exist. The document that comes closes is perhaps Demailly's notes on applications to algebraic geometry, again available on his website.

You should probably read, or rather violently leaf through, all this all at the same time, and then go bother the local complex geometry types with silly questions. There's no better way to learn a new language than to immerse oneself in it and try to talk to the natives.

[edit:] I see I forgot to mention two things: 1. Griffiths and Harris talk about residues in their book, and 2. "quadratic differentials" is secret code for "deformation theory on Riemann surfaces". There is no introduction for beginners to deformation theory. Anyone who says otherwise is either lying or severly underestimating the technical difficulties involved.

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thanks a lot ! I think that covers the complex geometry part ! –  Glougloubarbaki Nov 21 '11 at 21:20
    
I don't really have any ideas for the potential theory part, except for the obvious remark that Chern's "Complex manifolds without potential theory" will not at all be what you're looking for. –  Gunnar Magnusson Nov 21 '11 at 21:56

The standard textbook now on the subject is Complex Geometry:An Introduction by Daniel Huybrechts. It has relatively little in the way of prerequisites: A good course in complex analysis, basic differential topology and algebra. It looks excellent and doesn't try to do too much so it becomes overwhelming-this is a difficult subject after all.

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Thanks a lot for the helpful answer, this question is useful to me too. Wish to ask one thing : I wish to learn various aspects of Complex geometry and know of the books by Huybretchs,R O Wells, Voisin, Griffiths-Harris. Could you please advise which one or two of these to study so that the intersection between the selected books is minimal and the union maximal ? In terms of coverage which of these books deal with same aspects of the subject ? I hope leaving this comment and not asking a separate question in this regard is OK. –  user90041 Nov 10 '13 at 14:28
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@user90041 Griffiths-Harris is one of those ridiculously comprehensive old classics like Dunford-Schwartz or Eisenbud. It's a book you can learn an ENORMOUS amount from,but a lot of it is old-fashioned.I'd be very selective with it. Wells is kind of idiosyncratic and confuses people.(continued) –  Mathemagician1234 Nov 10 '13 at 20:08
    
@user90041(continued) The best and most complete "book" on the subject hasn't been published yet: It's Demailly's Complex Analytic and Differential Geometry, which has been available in draft form at his website for several years now with occasional revisions.I think you'll find it very helpful for your needs,it can be found here: www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf –  Mathemagician1234 Nov 10 '13 at 20:08
    
Thanks, it would be very nice of you to add your feedback about Voisin vs Huybretchs. Which one would you recommend choosing ? Do they cover similar stuff ? –  user90041 Nov 11 '13 at 1:59

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