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I am trying to get a good book that explains the Dolbeault Cohomology, does anyone know of a good one?

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up vote 6 down vote accepted

Chern's book "Complex manifolds without potential theory" is a good book, and it does explain Dolbeault Cohomology. But it's a short book, and it explains it concisely. If you need more details, you could also try Griffiths-Harris (but I greatly prefer Chern's book).

Kodaira's book "Complex manifolds and deformations of complex structures" is much more leisurely, and with great attention paid to exposition and detail (it doesn't appeal to some people, but I enjoyed it a lot).

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Chern's book is great. The only thing is that I never did find what " potential theory" was :) It's probably the only mathematics book I know of, whose title contains a concept (here "potential theory") which is not mentioned AT ALL in the rest of the book! – José Figueroa-O'Farrill Nov 11 '09 at 18:26
The funny thing is that Chern actually does discuss how to derive the Kahler form on a Kahler manifold from a function (i.e. its "Kahler potential"). Honestly, I also have no idea what "without potential theory" means in the title. :) – Danny Calegari Nov 11 '09 at 18:31
I am one of those people who really dislike Kodaira's book. One reason why I don't like it is because it does a lot of the theory (Kodaira-Spencer deformation theory) using coordinates and Cech cohomology. I think the theory is much clearer and cleaner if you do it without coordinates. – Kevin H. Lin Nov 11 '09 at 21:57
I think that's an entirely legitimate point of view, but it might also reflect your interests and taste. If you are interested in deformations of compact complex manifolds eg. as an example in the theory of elliptic PDE, you might appreciate expressing everything in terms of coordinates, since that makes it easier to adapt standard analytic tools (in which everything takes place in linear spaces). – Danny Calegari Nov 11 '09 at 23:50

I like "Hodge Theory and Complex Algebraic Geometry" by Voisin. The focus is on the Kahler case, but the early explanations of Dolbeaut cohomology are for all complex manifolds.

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If I remember right there should be something in "Differential Analysis on Complex Manifolds" by Wells.

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Another book that covers Dolbeault cohomology very nicely is "Complex Geometry" by Daniel Huybrechts. I highly recommend it.

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Chapter 0.2 of Griffith and Harris

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The references given in the wikipedia article .

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