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Can anyone recommend a good comprehensive introduction to functions of several complex variables that a) is fairly up to date, b) isn't a geometry or an algebra book only, but takes multiple viewpoints? I don't mind texts that require a lot of background as long as they present it intelligibly.

Just as good would be links to lecture courses by experts on the subjects that are accessible and do the job.

I've seen Griffith/Harris, Grauert and Gunning's 3 volume treatise. Gunning looks the best of the three, but I was hoping for something more up-to-date. This is still a very active field. I haven't seen Krantz yet.

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    $\begingroup$ I rather like Krantz's book. But I am not quite sure about the "takes multiple viewpoints" part of your requirement. $\endgroup$ – Willie Wong Jul 11 '10 at 13:12
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    $\begingroup$ Well,the objects of functions of several complex variables are manifolds with a complex topological vector space structure.Therefore,they are the centerpieces of the bulk of post-19th century analysis and geometry and the tools of sheaf theory via commutative algebra are deeply interwoven in them.As a result of all this,any "pure" approach-say,emphasizing analysis-only tells part of the story. $\endgroup$ – The Mathemagician Jul 11 '10 at 15:49
  • $\begingroup$ There's actually two approaches to the subject. One is analytical. Hoermander's book is the best reference I know. There are also good notes by Demailly on the d-bar problem and the Levi problem. Then there is the Oka-Cartan approach developed by Grauert and Remmert using algebra and sheaf theory. Gunning and Rossi "analytic functions" follows this approach. $\endgroup$ – Craig May 18 at 14:17
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I know this is somewhat late answer for the original question, but here goes. I've taught Several Complex Variables twice at Oklahoma State, and I wrote a hopefully easy to read textbook for the purpose. It is freely available online under a CC license:

http://www.jirka.org/scv/

It is not a reference work, it is a hopefully gentle introduction, something that would be covered in one semester by a beginning graduate student after at least one semester of one complex variable. It doesn't cover anything too deeply, but I did try to cover several different viewpoints rather than focus on one specific thing.

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  • $\begingroup$ It's great that your books are under the CC-BY-SA license. $\endgroup$ – Ben Crowell May 6 '18 at 22:39
  • $\begingroup$ +1.This is definitely a difficult but very important subject in both analysis and geometry that needs more user-friendly introductions to motivate young students to begin study, Thanks for making a contribution to that! $\endgroup$ – The Mathemagician May 21 at 23:58
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I really like

Several complex variables with connections to algebraic geometry and Lie groups By Joseph L. Taylor

http://books.google.co.uk/books?id=i8lUNpZ379MC&printsec=frontcover&dq=joseph+taylor+several+complex&source=bl&ots=7MVqaXprex&sig=o63JAEz_sKc-2e63qxcSsIBYwcQ&hl=en

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For a great introduction, try Raghavan Narasimhan, "Several complex variables", Chicago lectures in mathematics.

Well-written, concise and accessible.

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