Algebraic Geometry over the Complex Numbers

Complex algebraic varieties are sets of solutions of polynomial equations over the complex field. In algebraic geometry these varieties are studied using algebraic methods, usually from commutative and homological algebra and category theory. However, these varieties could also be studied with methods of complex differential geometry and algebraic topology. This two-sided approach enriches the subject and at the same time makes it harder to be learned by the beginner.

On the algebraic side of the subject there are many outstanding expositions, from classical approaches such as Shafarevich’s Basic Algebraic Geometry (Springer, first edition 1974 and in a two volume set: Volume 1 and Volume 2, 1994) to more advanced treatments such as Q. Liu’s Algebraic Geometry and Arithmetic Curves (Oxford, 2006) or Hartshorne’s Algebraic Geometry (Springer, 1977). On the transcendental side there are fewer expositions, the classical one being Principles of Algebraic Geometry, by P. Griffiths and J. Harris (Wiley, 1978) and a recent one is Huybrechts’s Complex Geometry (Springer, 2005). The book under review is a welcome addition to the literature on complex algebraic geometry. The approach chosen by the author balances the algebraic and transcendental approaches and unifies them by using sheaf theoretical methods.

After an introductory chapter with concrete examples of complex algebraic curves, the book properly starts in Part II. Here, sheaves of functions on an arbitrary topological space are introduced and then used to define and study complex manifolds and algebraic varieties in a unified way. Sheaf cohomology is then put to work in the special case of the de Rham cohomology of complex manifolds, proving the main topological results: the Künneth formula, Poincaré duality and the Lefchetz trace formula. These methods and results are then applied to the classical one-dimensional case. The last chapter of this part introduces simplicial methods to give realizations for some of the cohomology theories previously introduced: simplicial, singular and Čech cohomologies.

Part III is devoted to the transcendental approach. It starts with a proof of Hodge’s theorem (on a compact oriented complex manifold equipped with a metric, every de Rham cohomology class has a unique representative that minimizes the norm) using the heat equation. An immediate consequence is another proof of Poincaré duality in a strengthened form.

The main objective of this part, the interaction of de Rham and Hodge’s theories with the holomorphic structure on a complex manifold is addressed in the remaining chapters: Kähler manifolds and the Hodge decomposition are treated in Chapter 10. Chapter 11 treats the concrete case of compact complex surfaces, including the main results: the Neron-Severi group and the Riemann-Roch and Hodge index theorems. The remaining chapters go deeper into the subject, from the Hodge decomposition theorem and the homological methods needed to establish it to the hard Lefschetz theorem.

Part IV is devoted to the cohomology of coherent sheaves to prove Serre’s GAGA theorems that vastly generalize Chow’s theorem (every compact complex submanifold of a projective space is a nonsingular projective algebraic variety). This is then used to compute some invariants (the Hodge numbers) of projective spaces, hypersurfaces and double covers. The last chapter of Part IV is devoted to prove the Kodaira-Spencer theorem that the Hodge numbers of a complex variety do not change under deformation.

Part V, a single chapter, offers a glimpse of the many important conjectures that remain unproved so far, the so-called Grothendieck standard conjectures and their relation to the Hodge conjecture.

This is a well-written text, where intuition and rigor meet at some middle ground, with plenty of examples to illustrate the ideas being discussed. The prerequisites for reading the book are kept a minimum, however some previous exposure to elementary algebraic geometry on the level of Hulek’s Elementary Algebraic Geometry (AMS, 2003) and an introduction to calculus on manifolds as in Spivak’s Calculus on Manifolds (Harper and Collins, 1965) would certainly be helpful.

Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is fz@xanum.uam.mx.