References for complex analytic geometry? - MathOverflow

References for complex analytic geometry?

2010-05-30T12:59:29Z

I'm looking for references on the "algebraic geometry" side of complex analytis, i.e. on complex spaces, morphisms of those spaces, coherent sheaves, flat morphisms, direct image sheaves etc. A textbook would be nice, but every little helps.

Grauert and Remmert's "Coherent analytic sheaves" seems to contain what I want, but it is very dense reading. You could say I'm looking for sources to read on the side as I work through G&R, to get different points of views and examples. For example, B. and L. Kaup's "Holomorphic functions of several variables" talks about the basics of complex analytic geometry, but doesn't go into much detail.

My motivation is twofold. First, I'm studying deformation theory, which necessarily makes use of complex spaces, both as moduli spaces and objects of deformations, so while I can avoid using complex spaces at the moment they're certain to come in handy later. Second, I want to be able to talk to the algebraic geometers in my lab, so I should know what their schemes and morphisms translate to in the analytic case. I like reading as much as I can about what I'm trying to learn, so:

Do you know of other sources (anything: textbooks, lecture notes, survey articles, historical overviews, comparisons with algebraic geometry ...) that talk about complex spaces and their geometry?

Answer by Benoît Kloeckner for References for complex analytic geometry?

2010-05-30T13:15:28Z

When I needed to understand a little bit of complex algebraic geometry to study a complexe geometry problem, I used Griffith & Harris' book. It was quite easy to learn and extract just the informations I needed.

Answer by Daniel Larsson for References for complex analytic geometry?

2010-05-30T15:49:44Z

Since your interest is in deformation theory I would advise you to have a look at

"Introduction to singularities and deformations" by Greul, Lossen and Shushtin.

The first part of the book treats complex analytic geometry (complex space germs) and the second their deformation theory.

There's also a survey paper by Palamodov "Deformations of complex spaces" in Encyclopedia of Mathematics (Springer) which treats some foundational material as well.

Good luck!

Answer by Charles Siegel for References for complex analytic geometry?

2010-05-30T16:03:07Z

For deformation theory and complex manifolds, I'm a fan of Manetti's lecture notes.

Answer by Tony Pantev for References for complex analytic geometry?

2010-05-30T18:06:35Z

Two books that I like a lot:

1) Joseph Taylor's Several complex variables with connections to algebraic geometry and Lie groups .

2) Constantin Banica and Octavian Stanasila's "Algebraic methods in the global theory of complex spaces" , Wiley (1976)

Answer by Andrew L for References for complex analytic geometry?

2010-05-30T20:37:24Z

For complex geometry,which really is fundamental in analytic deformation theory,I strongly suggest 2 sources besides the classical source by Griffiths and Harris: Complex Geometry:An Introduction by Daniel Huybrechts,which has rapidly become the standard text on the subject,and the online text draft of a comprehensive work by Demially. The Demailly text is much more comprehensive and more advanced,with an emphasis on algebraic and differential geometry.But you may find it more helpful as it contains a great deal more near the research level. It can be found here: http://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf