3
$\begingroup$

Hello

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

Improve this question
$\endgroup$

6 Answers 6

Reset to default
7
$\begingroup$

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).

Improve this answer
$\endgroup$
4
  • 1
    $\begingroup$ 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! $\endgroup$
    – José Figueroa-O'Farrill
    Commented Nov 11, 2009 at 18:26
  • $\begingroup$ 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. :) $\endgroup$
    – Danny Calegari
    Commented Nov 11, 2009 at 18:31
  • $\begingroup$ 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. $\endgroup$
    – Kevin H. Lin
    Commented Nov 11, 2009 at 21:57
  • 1
    $\begingroup$ 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). $\endgroup$
    – Danny Calegari
    Commented Nov 11, 2009 at 23:50
5
$\begingroup$

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.

Improve this answer
$\endgroup$
3
$\begingroup$

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

Improve this answer
$\endgroup$
2
$\begingroup$

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

Improve this answer
$\endgroup$
1
$\begingroup$

Chapter 0.2 of Griffith and Harris

Improve this answer
$\endgroup$
0
$\begingroup$

The references given in the wikipedia article .

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .