Principles of Algebraic Geometry

Rating:
Summary: A review from a graduate student
Review: If you are a graduate student in mathematics or related fields and you are interested in learning algebraic geometry in the Griffiths-Harris way, then I suggest before buying this book to have a good background in the following:

1. Complex Analysis
2. Differential Geometry and calculus on manifolds
3. Homology-Cohomology Theory
4. Undergraduate Algebraic Geometry

Do not expect chapter 0, "Foundational Material", to be the place where you are supposed to build your "foundation". You can try the books of Michael Spivak, David A. Cox, Fangyang Zheng, among other books for foundational material but not chapter 0.

However, if you have most of the above-mentioned foundational material, then this book is good in presenting complex manifolds for example in chapter 0 section 2 and also in presenting (complex) holomorphic vector bundles, as well as many other things.

So, in summary, I would say a good book but not for students trying to learn the basics in algebraic geometry.

Rating:
Summary: The Book
Review: Just wanted to add the following:

1) The mathematics in this book is some of the most beautiful stuff I've ever seen. I don't in any way mean to deny the beauty of the Spec of a Ring, but - even if you have always planned on working in Grothendeick's world - I think this is worth reading for any algebraic geometer (regardless of what field you're living over).

With their bare hands, Griffiths and Harris prove some of the greatest results in maths. I learned more reading Chapter O than I did taking the entire collection of "first- year" grad courses (algebra & analysis). The material was more interesting, and it tied together in a way that had you remember all of it. From elliptic operator theory to the representation of sl(2), in the same chapter!

2) For string theorists trying to learn some of the math lingo, this is a necessary first step, though I would also highly recommend Candelas's notes, and Aspinwall's great paper, "K3 Surfaces and String Duality". Also, Brian Greene's notes are very nice. T. Hubsch's book is also great for the big picture, but I was disappointed by several non-trivial errors in his explanations of math concepts. I recommend all of the above to mathematicians as well - I am a mathematician, and I learned a lot of valuable side material from these physics sources. Especially in trying to understand mirror symmetry. Of course, Cox and Katz's newish book is also excellent for this.

3) My favorite parts: chap 1: divisors and line bundles, the exp sheaf sequence. read this, and then skip to the same picture for line bundles on a torus. the same type of bouncing back and forth works for getting the analogs between Reimann surfaces and complex surfaces...

actually, every page of this huge book has something valuable. I can't imagine what it was like to learn this field before this book came along. The price is exorbitant, but in the grand scheme of things, I've spent hundreds (thousands?) on math books that lie on my shelf, never to be explored. this one has given me years of enjoyment.

Rating:
Summary: Will never collect dust
Review: Once thought to be highly esoteric and useless by those interested in applications, algebraic geometry has literally taken the world by storm. Indeed, coding theory, cryptography, steganography, computer graphics, control theory, and artificial intelligence are just a few of the areas that are now making heavy use of algebraic geometry.

This book would probably be one the most useful one for those interested in applications, for it is an overview of algebraic geometry from the complex analytic point of view, and complex analysis is a subject that most engineers and scientists have had to learn at some point in their careers. But one must not think that this book is entirely concrete in its content. There are many places where the authors discuss concepts that are very abstract, particularly the discussion of sheaf theory, and this might make its reading difficult. The complex analytic point of view however is the best way of learning the material from a practical point of view, and mastery of this book will pave the way for indulging oneself in its many applications.

Algebraic geometry is an exciting subject, but one must master some background material before beginning a study of it. This is done in the initial part of the book (Part 0), wherein the reader will find an overview of harmonic analysis (potential theory) and Kahler geometry in the context of compact complex manifolds. Readers first encountering Kahler geometry should just view it as a generalization of Euclidean geometry in a complex setting. Indeed, the so-called Kahler condition is nothing other than an approximation of the Euclidean metric to order 2 at each point.

The authors choose to introduce algebraic varieties in a projective space setting in chapter 1, i.e. they are the set of complex zeros of homogeneous polynomials in projective space. The absence of a global holomorphic function for a compact complex manifold motivates a study of meromorphic functions and divisors. Divisors are introduced as formal sums of irreducible analytic hypersurfaces, but they are related to the defining functions for these hypersurfaces also, via the poles and zeros of meromorphic functions. For the mathematical purist, a "sheafified" version of divisors is also outlined. Divisors and line bundles are basically "linear" tools used to investigate complex varieties through their representation as complex submanifolds of projective space. In addition, various approaches are used to study codimension-one subvarieties, such as the results of Kodaira and Spencer. Although the famous Kodaira vanishing theorem is clothed in the language of Cech cohomology, this cohomology is represented by harmonic forms, thus making its understanding more accessible. The authors also show explicitly to what extent an algebraic variety can be thought of as a compact complex manifold via the Kodaira embedding theorem. Projective space of course is not the most complicated of constructions, as readers familiar with the theory of vector bundles will know. Grassmannians are an example of this, and they are introduced and discussed in the book as generalizations of projective space. And, just as in the ordinary theory of vector bundles, the authors show how to use Grassmannians to act as universal bundles for holomorphic vector bundles.

The presence of meromorphic functions will alert the astute reader as to the role of Riemann surfaces in the study of complex algebraic varieties. Indeed, in chapter 2, the authors cast many classical complex analytic results to modern ones, and they prove the famous Riemann-Roch theorem, which essentially counts the number of meromorphic functions on a Riemann surface of genus g. The theory of Abelian varieties is outlined, and the reader gets a taste of "Italian" algebraic geometry but done in the rigorous setting of Plucker formulas and coordinates.

Chapter 3 is a summary of some of the other methodologies and techniques used to study general analytic varieties, the first of these being the theory of currents, i.e differential forms with distribution coefficients. It is perhaps not surprising to see this applied here, given that it can handle both the smooth and piecewise smooth chains simultaneously. The currents are associated to analytic varieties and allow a definition of their intersection numbers and a proof that they are positive. The all-important Chern classes are introduced here, and it is shown that the Chern classes of a holomorphic vector bundle over an algebraic variety are fundamental classes of algebraic cycles. Most importantly the authors introduce spectral sequences, a topic that is usually formidable for newcomers to algebraic geometry.

The study of surfaces is studied in chapter 4, with the differences between its study and the theory of curves (Riemann surfaces) emphasized. The reader gets a first crack at the notion of a rational map, and the birational classification of surfaces is shown. Intuitively, one expects that the classification of surfaces would be easy if it were not for "singular points", and this is born out in the use of blowing up singularities in this chapter. Rational surfaces are characterized using Noether's lemma, and a rather detailed discussion is given of surfaces that are not rational, giving the reader more examples of rigorous "Italian" geometry.

Rating:
Summary: algebraic geometry: the real stuff
Review: The book is beautifully written and easy to read, with emphasis on geometric picture instead of abstract nonsense. By far the best introduction to algebraic geometry for string theorists.

Rating:
Summary: More geometry, less algebra.
Review: This book is a throwback to the time when algebraic geometry was a branch of geometry rather than category theory. As wonderful as the books by Mumford and Hartshorne are, they are rather long on abstract nonsense and short on geometry. This book is a refreshing exception to the 'modern' trend. Actually, there is a renaissance in applications of algebraic geometry to surprizing fields such as encryption and string field theory, and these are more in the spirit of this book than those of the Grothendieck school. Except for the obscenely high price and occasional typos, I highly recommend this book, especially to geometrically inclined mathematicians who don't really care about the category of schemes over an arbitrary field

Rating:
Summary: Absolutely indispensable
Review: This book is fabulous - it is an indispensable reference for complex algebraic geometry. It is very clearly written and ideas are always motivated by examples and problems. Moreover, if you want to learn modern algebraic geometry, it's imperative to learn the classical case (over the complexes - which in practice is easier to work in) in order to understand the generalisations a la Grothendieck.

Rating:
Summary: Work of Art
Review: This is an amazing book with an amazing subject (complex algebraic geometry). Every section presents something interesting and wonderful. I've only read chapters 0 (Complex manifolds, Hodge theory), 1 (Divisors & line bundles, vanishing theorems, embeddings), and 2 (Riemann surfaces). I had had a bad experience with alg geom before this book. Required reading for mathematicians in complex manifolds, algebraic geometry, or string theorists. There are some very trivial typos scattered, but nothing problematic in the least (like capital lambda instead of a big wedge, or indices). If you read the book carefully you will get a lot out of it.

Rating:
Summary: Work of Art
Review: This is an amazing book with an amazing subject (complex algebraic geometry). Every section presents something interesting and wonderful. I've only read chapters 0 (Complex manifolds, Hodge theory), 1 (Divisors & line bundles, vanishing theorems, embeddings), and 2 (Riemann surfaces). I had had a bad experience with alg geom before this book. Required reading for mathematicians in complex manifolds, algebraic geometry, or string theorists. There are some very trivial typos scattered, but nothing problematic in the least (like capital lambda instead of a big wedge, or indices). If you read the book carefully you will get a lot out of it.