I wish to learn Complex Geometry and am aware of the following books : Huybretchs, Voisin, Griffths-Harris, R O Wells, Demailly. But I am not sure which one or two to choose. I am interested in learning complex analytic & complex algberaic geometry both.

Could somebody please advise me which of these books deal with the same or similar aspects of the subject ? If I am not mistaken, Huybretchs and Voisin deal with similar aspects ? and Huybretchs would be relatively elementary or may be a preparation for Voisin ?

I would like to select two books (preferably from above-listed ones, though other suggestions are welcome) such that the intersection between their contents is minimal and the union maximal.

Any comments about the above-mentioned books will be very helpful.

up vote 18 down vote accepted

Well, you'll really want to read them all at some point. To start with, take Griffiths-Harris for geometric insight and Huybrecths for company (his chapter 1.2 is amazing). Voisin is very good and at first covers the same ground as Huybrecths, but is more advanced (do read the introduction to Voisin's book early, it sets the scene quite well). Demailly's book is where all the details are, you'll want that one for proofs of the main theorems like Hodge decomposition, Kodaira vanishing etc. There's also a new book by Arapura that looks very user-friendly.

And now for some clearly false generalities: The books by Huybrechts, Voisin and Arapura have very algebraic points of view; they were written by people who are mainly algebraic geometers and (to simplify greatly) think in Spec of rings. By contrast, Demailly and Griffiths-Harris have more differential-geometric points of view and use metrics and positivity of curvature as their main tools. I'll take the opportunity to also recommend Zheng's wonderful "Complex differential geometry" for an alternative introduction to that point of view. You'll need to know how to use all of these tools (as do all those people, of course).

So, to sum things up: $$ \begin{array}{ccc} & \hbox{introduction} & \hbox{advanced} \\ \hbox{algebraic} & \hbox{Arapura, Huybrechts} & \hbox{Voisin}\\ \hbox{metric} & \hbox{Griffiths-Harris, Zheng} & \hbox{Demailly} \\ \end{array} $$

  • Thanks a lot for the answer, this is indeed very helpful and informative. Just one more question : Let us say I plan that I shall read Voisin $ Demailly. What are the prerequisites ? I know basic Commutative and Homological algebra & Complex Analysis of one variable (Lang, Stein-Shakarchi). But otherwise my knowledge of analysis is limited to what is done in the book Principles of Mathematical Analysis by Rudin. Is this sufficient preparation to read Demailly ? (I should mention that I know Manifold theory & Differential Geometry (forms, de-Rham theory, connections and curvature on bundles)) – user90041 Nov 20 '13 at 4:59
  • Yeah, that sounds all right. If you need anything else you'll pick it up along the way. On the metric front you'll end up doing a surprising amount of linear algebra, but the only hardcore things needed are functional analysis techniques for proofs of some of the heavier theorems, like Cartan-Serre finiteness and Hodge isomorphism & decomposition. You can take those on faith for now or read up on the necessary background when you need it. – Gunnar Þór Magnússon Nov 20 '13 at 13:57
  • Gunnar, thank you for a very nice post. Can I ask you the following questioin: how valuable will the reading of Griffiths-Harris be if I have already read Huybrechts and vol.1 of Voisin (I am also familiar with shemes)? The thickness of GH is a little discouraging (even for someone who had persisted through Vakil's notes last year). And do people actually read it as a textbook or rather use it as a reference monongraph? – Bananeen Nov 12 '17 at 23:26

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