## References for complex analytic geometry?

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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?

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The G&R book doesn't really address flatness (but handles everything else); they use a notion of "active germs" (non-zero-divisors in a suitable sense) to get around it. The series by Gunning & Rossi treats the case of analytic spaces which are reduced. Assuming reducedness is rather restrictive (e.g., cannot make fibers products, even for analytic fibers of a branched covering of Riemann surfaces), but it may be a good way of easing into the general case and becoming more comfortable with sheaf-theoretic reasoning.  – BCnrd May 30 2010 at 13:30
Also, the old Seminaire Cartan lectures by C. Houzel on analytic spaces give a nice treatment of the local structure of analytic spaces and properties of local rings on them, especially the henselian property of the local rings and local structure of analytic maps with isolated point in a fiber. You may find that a useful prelude to Grauert-Remmert (if not already covered by Gunning & Rossi).  – BCnrd May 30 2010 at 13:50
Thanks for the tips, I'll look both Gunning & Rossi and Houzel up tomorrow. Some random googling also turned up "Complex analytic geometry" by G. Fischer in Lecture notes in Mathematics. The index is available on SpringerLink (springerlink.com/content/l37102231p72/…), it looks like it talks about similar things as Grauert and Remmert do. – Gunnar Magnusson May 30 2010 at 18:22

## 5 Answers

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)

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Tony, the B&S book is certainly the ultimate reference post-Grauert-Remmert for "algebraic" aspects (flatness, $S_n$ loci, etc.). I didn't know that it was published in a language other than French & Romanian.  – BCnrd May 30 2010 at 22:50
It exists in English - published by John Wiley and Sons in 1976. Unfortunately our library doesn't have it but one can find it through inter library loan. – Tony Pantev May 31 2010 at 2:26
Excellent! Thank you Tony, references like these are exactly what I was hoping for. – Gunnar Magnusson May 31 2010 at 10:58

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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

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Huybrechts' book is very good indeed, but like Griffiths and Harris almost exclusively discusses the manifold case. Demailly is my thesis advisor so I'm pretty familiar with his book. He does go into more detail, especially in Chapters 2 and 4, and eventually 9 when he'll have time to finish writing it, but his focus really is on the smooth case, the complex spaces being somewhat treated as digressions. And since we're on the subject of books on smooth complex manifolds, Complex differential geometry by Fangyang Zheng is an absolute dream. – Gunnar Magnusson May 31 2010 at 10:49

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.

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Thanks Benoit. Griffiths and Harris is very nice, but treats almost exclusively complex manifolds (they do mention analytic sets at the beginning). I'm looking for something more like an analytic version of Harthshorne, where the spaces in question are ringed spaces locally isomorphic to analytic sets. – Gunnar Magnusson May 30 2010 at 13:39

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!

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Thanks Daniel, I'll take a look at both of those. Is the Palamodov paper the same one as appeared in "Russian Mathematical Surveys" in 1976? I've been looking for that one, but it's hard to find copies of that journal. Incidentally, there's a huge list of references and a historical overview of deformation theory by Doran here: www.math.columbia.edu/~doran/Hist%20Ann%20Bib.pdf - It's been extremely useful for finding insightful papers and articles. – Gunnar Magnusson May 30 2010 at 18:07

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

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I actually have those on my desk. The examples are very nice, and the historical surveys are a good touch. – Gunnar Magnusson May 30 2010 at 16:28
I like these notes a lot, too. If you like Manetti's notes, I recommend also looking at Kontsevich's and Gerstenhaber's work on deformation theory for a more general picture. – Kevin Lin May 30 2010 at 19:27
I think at some point,I'm going to begin a thread on general deformation theory where we need to distinguish carefully between algebraic and analytic deformation theory,as well as between classical algebraic deformation theory (Gerstenhaber,Schessinger) and the modern theory (Harsthorne,Lurie). They all appear quite different,but they are clearly connected. Those interconnections aren't really clear to the beginner or even someone with a classical background like me.  – Andrew L May 30 2010 at 20:29
Andrew, have you seen mathoverflow.net/questions/385/… ? I believe that differential graded Lie algebras are the bridge between the "classical" and the "modern" theory. – Kevin Lin May 31 2010 at 5:30
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