Principles of Algebraic Geometry (Pure and Applied Mathematics) 1st Edition by Phillip Griffiths (PDF)

Description

A comprehensive, self-contained treatment presenting general results of the theory. Establishes a geometric intuition and a working facility with specific geometric practices. Emphasizes applications through the study of interesting examples and the development of computational tools. Coverage ranges from analytic to geometric. Treats basic techniques and results of complex manifold theory, focusing on results applicable to projective varieties, and includes discussion of the theory of Riemann surfaces and algebraic curves, algebraic surfaces and the quadric line complex as well as special topics in complex manifolds.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐I’m just starting to read this book, but everything I’ve read is very interesting. Griffiths and Harris contrasts with many of the modern abstract approaches to algebraic geometry (i.e. Hartshorne) in how concrete and hands on it is. There are coordinates and formulas and integrals. These are the types of proofs that make sense to me, and the theorems are beautiful.It’s not just a reference book; I find a good introduction to complex algebraic geometry. But it’s not for the faint of heart. I’ve spent a lot of time with Huybrecht’s Complex Geometry which covers a lot of the material in chapter 0. I think these books complement each other well.A caveat is that the printing is poor, and the words are blurry. Wiley needs to step up its game!

⭐Advance and comprehensive. A good reference book.

⭐Very Good!

⭐Excellenta book

⭐thank you

⭐A classic

⭐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 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.

⭐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 Analysis2. Differential Geometry and calculus on manifolds3. Homology-Cohomology Theory4. Undergraduate Algebraic GeometryDo 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.

⭐This is still one of the best references available for complex geometers (either for novices or specialists)As many others pointed out already, it has several typos and innacuracies sprinkled throughout the text (from harmless symbol omissions to wrong definitions or proofs). However, this text is still unmatched in its scope and breadth (perhaps Demailly’s notes can be considered a worthy successor/companion stressing the analytic point of view)All in all, this venerable text has stood the test of time and surely will remain a mandatory read for the upcoming generations of complex geometers.If you are a differential geometer or an abstract algebraic geometer you might have heard of this book already; don’t fear it, it has a lot of beautiful mathematics. Read it, you won’t regret it , just be sure to read it alongside some of the other great contemporary texts (e.g. Huybrechts, Voisin, Demailly, Lesfari, etc.)

⭐Studio geometria algebricaOn m’a supprimé mon commentaire, qui n’etait pas agressif mais relevait de mon expérience avec ce vendeur, Juste la vérité….qui n’est pas appréciée hélas !Pourtant le futur acheteur doit être informé..!Donc ne rien dire qui dérange….Good.

⭐good

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