Deformation theory is a ubiquitous subject: From the Taylor expansion in Calculus to the deformation of Galois representations. In its modern form, deformation theory began with the work of Riemann. In a paper published in 1857 on the theory of Abelian functions, Riemann conjectured that the set Mg of non-isomorphic compact Riemann surfaces of genus g at least 2 can be parametrized by 3g–3 complex parameters, which he called “moduli.” Riemann’s arguments were formalized by Teichmüller in the 1930s, and in the next decades Ahlfors, Rauch and Bers proved that the set Mg has a natural complex structure, making it a complex manifold of dimension 3g–3. This moduli space is the first instance of a natural algebraic (complex) structure on a universal deformation space.
The second stage in these developments is the Kodaira-Spencer deformation theory of the complex structure of a compact complex manifold of arbitrary dimension. Kodaira and Spencer observed that since one such manifold X is defined by a set of coordinate charts, an infinitesimal deformation of X is a shifting of these charts. It then happens that one such infinitesimal deformation is represented by an element of the first cohomology group of X with coefficients in the sheaf of germs of holomorphic vector fields, and the obstruction to the extension of one such deformation lies in a second cohomology group. It can be shown that the Kodaira-Spencer theory, when applied to a one dimensional compact complex manifold X, i.e., a compact Riemann surface X, of genus g at least 2, recovers Riemann count 3g–3 as the dimension of the corresponding first cohomology group of X with coefficients in the sheaf of holomorphic vector bundles on X. Kodaira and Spencer proved all these results in a series of deep beautiful papers in the 1950s. A complete exposition, together with some historical commentaries, can be found in K. Kodaira’s Complex Manifods and Deformation of Complex Structures (Springer, 1986).
Grothendieck translated the work of Kodaira and Spencer to the setting of abstract algebraic geometry, developing a theory of first order deformations and cohomological obstructions that is applicable not only to smooth complex manifolds but to general schemes and analytic spaces. The notion of a moduli space is then recovered in the formalism of representable functors. Most of Grothendieck’s results are in his exposés in the Bourbaki seminar and are collected in the FGA (Fondements de la géométrie algébrique, 1957–1952; see also Fundamental Algebraic Geometry: Grothendieck's FGA Explained). M. Schlesinger extended Grothendieck’s formalism of representability to functors on the category of complete Noetherian local rings and proved a general criterion for pro-representability of these functors. There are immediate applications to infinitesimal deformations and Picard functors, which are important since, in general, these are not representable by schemes, but by M. Artin’s algebraic spaces. Later on, Schlesinger and Lichtenbaum introduced in “The cotangent complex of a morphism” (Trans. A. M. S., 128 (1967), 41-70) a device to compute obstructions to infinitesimal deformations, which were further generalized by Grothendieck and L. Illusie.
The book under review, conceived as a textbook with exercises at the end of every section, gives an introduction to the basics of deformation theory, with plenty of examples to illustrate the techniques just introduced. The book focuses its attention on some standard situations, such as the deformation of subschemes of a fixed scheme. A standard example in this situation is the Hilbert functor that parametrizes closed subschemes of a given projective scheme. Another such situation is the classification of vector bundles on a fixed scheme. An example in this situation is the Picard functor that parametrizes invertible sheaves on a given scheme. One last situation being considered is the deformation of abstract schemes, including the local study of deformation of singularities and the global study of deformation of nonsingular varieties. An important example of this situation is the existence and properties of the moduli space (variety) of curves.
A quick overview of the contents of the book should be enough to give a sense of the well-organized material: Chapters 1 and 2 give an introduction to the theory of first order and higher order deformations and obstructions. In the first chapter, by considering deformations over the ring of dual numbers, one gets the so-called first order infinitesimal deformations, using as motivation the Hilbert scheme that describes families of closed subschemes of a projective space over a field.
For the first two situations described above, the usual cohomology of coherent sheaves suffices, but not for the deformation of abstract schemes and the moduli problem. A section in this chapter introduces the Ti functors of Schlesinger and Lichtenbaum that are needed in that case. In chapter 2, attention focuses on the obstruction theory for the extension of a deformation over a local Artin ring to a larger Artin ring. Several well-chosen examples illustrate the various situations where this obstruction theory can be applied. In chapter 3, by taking the limit over larger and large Artin rings we arrive at the notion of formal deformations, the best possible scenario being the case when the corresponding functor is pro-representable. This chapter includes Schlesinger’s criterion for the pro-representability of functors on Artin rings, with applications to the various situations described above. Examples include the local Hilbert and Picard functors. This chapter ends with a section on the question of algebraization of formal moduli.
The last chapter is devoted to some global questions. In particular, it treats the question of the existence of a global moduli space that should parametrize the classes of objects being considered. Here the author wisely avoids the construction or proof of existence of most general moduli spaces, instead focusing on a description of the properties that these constructions must have. Thus, we find in this chapter the notions of “coarse” and “fine” moduli spaces and a discussion of when a given functor is or not representable. As examples of these constructions, the author treats the moduli of elliptic curves in detail, sketches some properties of the moduli of curves of higher genus, and gives an introduction to Mumford’s notion of moduli of (stable or semistable) vector bundles on a given scheme.
Since deformation theory could be considered a central topic in algebraic geometry, the publication of a textbook where some of the main results and methods are collected in one place is certainly welcome. Moreover, the inclusion of exercises and plenty of examples, make this book suitable for a course on this topic or for self-study, with the only prerequisite the now standard textbook on Algebraic Geometry by the same author.
Finally, on the topic of deformation theory, I should mention the recent monograph Deformation of Algebraic Schemes (Springer, 2006) by E. Sernesi. The book under review and Sernesi’s monograph nicely complement each other.
Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is email@example.com.