Difference between several books on complex geometry

Question

I would like to learn some complex geometry, especially the interaction between algebraic geometry and complex geometry.

I found that there are several famous books:

• Huybrechts, Complex Geometry;
• Voisin, Hodge theory and complex algebraic geometry;
• Griffiths & Harris, Principles of algebraic geometry;
• Demailly, Complex analytic and differential geometry;
• Carlson, Muller-Stach, Peters, Period Mappings and Period Domains;
• Cattani, El Zein, etc., Hodge Theory.

I am not familiar with differential geometry. So I would like to start with the most basic (or self-contained) books, as well as the one can help me understand algebraic geometry. In order to choose the best book for me, I would like to know what these books mainly talk about, difference between them, and the roadmap between these books.

The only thing I know is, according to what Huybrechts said in preface, Voisin and GH's books can be regarded as further readings. However, since I am an entire layman to complex geometry, even after going through the tables of contents of both books, I still don't know if these two books are equivalent, or focus on different topics.

Any comments, reviews, and instruction are welcome! Thanks a lot!

Answer 1

I think the best option with little knowledge of differential geometry is definitely Huybrechts' Complex Geometry. His approach is to introduce just the right amount of differential geometry that one needs to understand the proofs of the big theorems. Plus, he mixes in the sheaf theory with the analytic theory in a nice way. It will definitely serve you well if you know at least a bit of de Rham cohomology before starting. If you go with this, don't be afraid to skip chapter 1.2 on first reading and refer back to it when you need it. That chapter has a fantastic treatment of the linear algebra needed for complex geometry, but you might feel more motivated to read it once you know how it will be used.

Voisin's book is a better second course in my opinion. This set of books goes much deeper into both the analysis and the homological algebra of the theory. I've found the proofs in that book to be of greater depth than Huybrechts'. Plus, no analysis punches are pulled, as opposed to Huybrechts' book. So, if you feel like you want to read the proof of the Hodge decomposition, Voisin's book has the detail you'll need.

Griffiths and Harris and Demailly's books are excellent from the analytic viewpoint, but without differential geometry background I would wait on these. They are both almost entirely differential geometric in nature. G&H's chapter 0 does have a ton of great background material for complex geometry that is nice to refer to on occasion, though.

Demailly's book is hard. It is great in that it really starts at the beginning with definitions of smooth manifolds. That being said, if heavy analysis isn't your cup of tea, then that book will be a tough read. I use it more as a reference when I need to dig deep into analytic details.

CMP's book is rather orthogonal to the rest on this list. If you're just trying to learn algebraic geometry, I would suggest reading the first chapter up until Hodge structures are introduced. The way they illustrate the case of elliptic curves and tori is a really nice intro to period domains/maps, but I personally found the next few chapters incomprehensible on first pass. I personally felt that the details were far too sparse in this book.