# Learning Complex Geometry - Textbook Recommendation Request

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.

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)) Nov 20, 2013 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. Nov 20, 2013 at 13:57

Having read considerable chunks from Huybrechts, Voisin, Griffiths-Harris and Demailly, I want to suggest a slightly different viewpoint on the relationship between these books that hopefully complements the brilliant answer by Gunnar Þór Magnússon.

Books by Huybrechts and Demailly focus on the theoretical foundation of complex geometry, with the book by Huybrechts being considerably more elementary than the course by Demailly (the latter touches on singular complex spaces, currents and spectral sequences, to name a few). I think it would be a true statement to say that Demailly fully covers the contents of Huybrecths' book, but then provides much more on top of that. In particular, the end of Demailly's course also taps into the research interests of Demailly himself and discusses notions of positivity for vector bundles and $$L^2$$ estimates. The very last section of the book proves Grauert's Direct Image Theorem, which in and of itself shows how much technique is developed in the book.

Voisin's books go much deeper into Hodge theory, just as the title suggests. The beginning of Volume I covers some of the same foundations as Huybrechts and Demailly up to the discussion of the Hodge decomposition in cohomology. However, starting with chapter 7 the book goes deeper into Hodge theory proper, and culminates in the discussion of variations of Hodge structures in chapters 9 and 10. The end of Volume I lays ground for the discussion of analytic and algebraic cycles in Volume II, again something that is mentioned in Demailly on only a couple of pages. Volume II then discusses advanced topics close to the research level in variations of Hodge structures (relevant for moduli theory), algebraic cycles and fine topology of algebraic varieties. In conclusion, only about a half of Volume I overlaps with Huybrechts/Demailly.

The book by Griffiths and Harris shines (in my opinion) when it comes to examples and applications of the foundational theory. The material of chapters 0 and 1 is fully covered by Huybrechts/Demailly, and in a more detailed fashion. But chapters 2 and 4 on curves and surfaces contain a wealth of beautiful geometric material that is scattered in the literature, but here you can find it all in one place. This "examples and applications" part overlaps neither with Huybrecths/Demailly nor with Voisin.

PS. I am not really sure whether there exists a better exposition for parts of chapters 3 and 5 of Griffiths-Harris. In case there does not, it should still be considered a go-to resource to learn about applications of currents and residues in complex geometry.