##
**Deformations of algebraic schemes.**
*(English)*
Zbl 1102.14001

Grundlehren der Mathematischen Wissenschaften 334. Berlin: Springer (ISBN 3-540-30608-0/hbk). xi, 339 p. (2006).

Deformation theory in algebraic geometry is a basic tool that is just as old as the idea of classifying algebraic varieties. In his famous memoir on abelian functions from 1857, Riemann initiated the study of deformations of the complex structure of compact one-dimensional complex manifolds of given topological genus \(g\), that is of compact Riemann surfaces (or non-singular complex projective curves) of genus \(g\). The deformation of complex algebraic surfaces seems to have been considered first by Max Noether in 1888. However, a systematic deformation theory for higher-dimensional complex manifolds could be developed only as late as in 1958 and in the years thereafter, thus even more than 100 years after Riemann’s pioneering memoir, and with the modern conceptual framework of sheaf theory, sheaf cohomology, and Hodge theory as a crucial ingredient [cf.: K. Kodaira, “Complex manifolds and deformation of complex structures.” Grundlehren der Mathematischen Wissenschaften, 283. New York etc.: Springer-Verlag. X, 465 (1986; Zbl 0581.32012)].

This modern approach to complex-analytic deformation theory is well-known as Kodaira-Nirenberg-Spencer-Kuranishi theory, and its immediate effect on the development of classification theory in algebraic geometry has been equally revolutionary. In fact, A. Grothendieck immediately realized the functorial character of the Kodaira-Spencer theory of infinitesimal deformations of complex-analytic manifolds, and he outlined an analogous approach within his newly created foundational framework of algebraic schemes and formal algebraic geometry right away. These fundamental ideas, which eventually led to what is now called algebraic deformation theory, were published in a series of Bourbaki seminar expositions collected in his celebrated “Fondements de la Géométric Algébrique” [A. Grothendieck “FGA”, Secretariat Math. Paris (1962; Zbl 0239.14002)]. In the sequel, algebraic deformation theory has rapidly grown into a vast and central topic in modern abstract algebraic geometry, due to its crucial importance in regard to variational problems, including local properties of moduli spaces of varieties, vector bundles, and singularities. In its present state of art, algebraic deformation theory is highly formalized, both conceptually and methodically rather involved, widely ramified within algebraic geometry, and therefore not easily accessible to non-experts in the field. Although being of such importance and ubiquity in variational algebraic geometry, and while still developing rapidly, algebraic deformation theory had so far not yet found an adequate reflection in the relevant textbook literature. This fact made it rather difficult for non-specialists to find a solid orientation in this vast field of contemporary mathematical research, all the more so as numerous subtle technicalities and allegedly well-known results are scattered in the huge literature as “folklore”, without rigorous and detailed proofs. The book under review is an attempt to partially fill this bothersome gap in the literature, and to provide a largely self-contained and comprehensive account of deformation theory in classical algebraic geometry, with complete proofs of those results and techniques that are needed to fully understand the local deformation theory of algebraic schemes over an algebraically closed field. This includes the careful explanation of those basic tools that are indispensable, for example, in the local study Hilbert schemes, Quot schemes, Picard schemes, and other classifying objects in this context. In this vein, and for the first time in the literature, the author compiles some of the many folklore results scattered in the literature, with detailed and systematic proofs, which must be seen as a just as valuable as rewarding contribution towards a solid foundation of algebraic deformation theory, and as an utmost useful service to the mathematical community likewise. As for the contents, the present book consists of four chapters, each of which is divided into several sections and subsections. In addition, and for the convenience of the non-expert reader, there are five appendices devoted to some basic facts from commutative algebra and algebraic geometry which are used throughout the text.

After a brief introduction, in which an outline of the complex-analytic infinitesimal deformation theory à la Kodaira-Nirenberg-Spencer-Kuranishi serves as an explanation of the logical structure of algebraic (and functorial) deformation theory, Chapter 1 introduces the reader to infinitesimal deformations in an elementary fashion. The first section discusses extensions of rings and algebras, together with their generalizations to schemes over a base scheme, whereas the second section explains locally trivial deformations of schemes. This includes infinitesimal deformations of non-singular affine schemes, automorphisms of deformations and their extendability properties, first-oder locally trivial deformations, higher-order deformations, and the related (cohomological) obstruction theory for deformations. Chapter 2 provides the foundations of formal deformation theory. Starting with the concepts of formal smoothness and relative obstruction spaces for ring extensions, the author reconsiders infinitesimal deformations of algebraic schemes via M. Schlessinger’s theory of functors of Artin rings, culminating in Schlessinger’s famous theorem on the existence of (semi-) universal formal deformations [cf.: M. Schlessinger, Trans. Am. Math. Soc. 130, 208–222 (1968; Zbl 0167.49503)]. This is followed by a discussion of deformation functors and local moduli functors in general, including their obstruction spaces as well as concrete applications to the deformation theory of algebraic surfaces in characteristic zero. In this regard, the author provides a largely self-contained treatment of formal deformation theory along the classical approach, that is without introducing cotangent complexes (à la L. Illusie) or methods of differential graded Lie algebras as more recent tools in deformation theory.

Chapter 3 illustrates the general theory of deformation functors by several concrete examples. With the single exception of deformations of algebraic vector bundles, which have already been exhaustively treated in several recent research monographs like the book of D. Huybrechts and M. Lehn [“The geometry of moduli spaces of sheaves”. Aspects Math. E 31 (1997; Zbl 0872.14002)], the author describes in great detail the most important deformation functors in current algebraic geometry, mainly by carefully verifying Schlessinger’s conditions for the existence of (semi-)universal deformation spaces in the respective cases. Moreover, the first-order deformations, i.e., the tangent spaces of these functors, as well as the corresponding obstruction spaces are thoroughly analyzed, thereby revealing a similar pattern in almost all of these exemplary cases. As for the concrete examples treated in this chapter, the author discusses the deformation functor of an affine scheme with at most quotient singularities, the local (relative) Hilbert functor of closed subschemes of a scheme, the local Picard functor of a scheme, deformations of sections of an invertible sheaf on a scheme, deformations of various types of morphisms of schemes, deformations of a closed embedding, and (co-)stable subschemes.

Chapter 4 gives a thorough introduction to Hilbert schemes, Quot schemes, and flag Hilbert schemes. As the author points out, these objects are needed to construct important examples of global deformations, and to study their local behaviour in the framework of the main theme of the present text. Besides, until now it was rather difficult to give precise references for many results on the geometry of these classifying objects, just like in local deformation theory, and this very fact has been another reason for the author to include that chapter in his book. Together with the very recent monograph “Fundamental Algebraic Geometry: Grothendieck’s FGA Explained” by B. Fantechi et al. [Math. Surv. Monogr. 123 (2005; Zbl 1085.14001)], this chapter in the book under review provides the only (and overdue) reasonably comprehensive exposition on Grothendieck’s Hilbert and Quot schemes. Apart from a general introduction to Castelnuovo-Mumford regularity, flattening stratifications, Hilbert schemes, Quot schemes, flag Hilbert schemes, and Grassmannians, together with applications to families of projective schemes, there is a concluding section on plane curves, their equisingular infinitesimal deformations, and their so-called Severi varieties. The author’s approach to the proof of existence of nodal curves with an arbitrary number of singularities is very original and apparently new. Based on the use of multiple point schemes, this example strikingly illustrates the power of algebraic deformation theory even in classical curve theory.

The five appendices at the end of the book collect some basic standard topics and are titled as follows: A. Flatness; B. Differentials; C. Smoothness; D. Complete intersections; E. Functorial language. Most of the results presented here come with full proofs, which further strengthens the already high degree of self-containedness of the book.

There are no explicit exercises or working problems accompanying the text, but there is a wealth of illustrating concrete examples, additional remarks, historical annotations, and hints to further reading. The bibliography includes 190 references, ranging from the very classical up to the most recent articles and books, and both a very carefully compiled list of used symbols and a just as thorough alphabetical index considerably enhance the value of the entire treatise.

Without any doubt, this is a masterly book on a highly advanced topic in algebraic geometry. The author’s style of writing captivates by its high degree of comprehensiveness, completeness, rigour, sytematical exposition, creative originality, lucidity, and user-friendliness in a like manner. The entire text is kept at a level that makes it suitable for graduate students with a solid background in commutative algebra, homological algebra, and basic algebraic geometry. But even for experts and active researchers in algebraic geometry, this unique book on algebraic deformation theory offers a great deal of inspiration and new insights, too, and its future role as a standard source and reference book in the field can surely be taken for granted from now on.

This modern approach to complex-analytic deformation theory is well-known as Kodaira-Nirenberg-Spencer-Kuranishi theory, and its immediate effect on the development of classification theory in algebraic geometry has been equally revolutionary. In fact, A. Grothendieck immediately realized the functorial character of the Kodaira-Spencer theory of infinitesimal deformations of complex-analytic manifolds, and he outlined an analogous approach within his newly created foundational framework of algebraic schemes and formal algebraic geometry right away. These fundamental ideas, which eventually led to what is now called algebraic deformation theory, were published in a series of Bourbaki seminar expositions collected in his celebrated “Fondements de la Géométric Algébrique” [A. Grothendieck “FGA”, Secretariat Math. Paris (1962; Zbl 0239.14002)]. In the sequel, algebraic deformation theory has rapidly grown into a vast and central topic in modern abstract algebraic geometry, due to its crucial importance in regard to variational problems, including local properties of moduli spaces of varieties, vector bundles, and singularities. In its present state of art, algebraic deformation theory is highly formalized, both conceptually and methodically rather involved, widely ramified within algebraic geometry, and therefore not easily accessible to non-experts in the field. Although being of such importance and ubiquity in variational algebraic geometry, and while still developing rapidly, algebraic deformation theory had so far not yet found an adequate reflection in the relevant textbook literature. This fact made it rather difficult for non-specialists to find a solid orientation in this vast field of contemporary mathematical research, all the more so as numerous subtle technicalities and allegedly well-known results are scattered in the huge literature as “folklore”, without rigorous and detailed proofs. The book under review is an attempt to partially fill this bothersome gap in the literature, and to provide a largely self-contained and comprehensive account of deformation theory in classical algebraic geometry, with complete proofs of those results and techniques that are needed to fully understand the local deformation theory of algebraic schemes over an algebraically closed field. This includes the careful explanation of those basic tools that are indispensable, for example, in the local study Hilbert schemes, Quot schemes, Picard schemes, and other classifying objects in this context. In this vein, and for the first time in the literature, the author compiles some of the many folklore results scattered in the literature, with detailed and systematic proofs, which must be seen as a just as valuable as rewarding contribution towards a solid foundation of algebraic deformation theory, and as an utmost useful service to the mathematical community likewise. As for the contents, the present book consists of four chapters, each of which is divided into several sections and subsections. In addition, and for the convenience of the non-expert reader, there are five appendices devoted to some basic facts from commutative algebra and algebraic geometry which are used throughout the text.

After a brief introduction, in which an outline of the complex-analytic infinitesimal deformation theory à la Kodaira-Nirenberg-Spencer-Kuranishi serves as an explanation of the logical structure of algebraic (and functorial) deformation theory, Chapter 1 introduces the reader to infinitesimal deformations in an elementary fashion. The first section discusses extensions of rings and algebras, together with their generalizations to schemes over a base scheme, whereas the second section explains locally trivial deformations of schemes. This includes infinitesimal deformations of non-singular affine schemes, automorphisms of deformations and their extendability properties, first-oder locally trivial deformations, higher-order deformations, and the related (cohomological) obstruction theory for deformations. Chapter 2 provides the foundations of formal deformation theory. Starting with the concepts of formal smoothness and relative obstruction spaces for ring extensions, the author reconsiders infinitesimal deformations of algebraic schemes via M. Schlessinger’s theory of functors of Artin rings, culminating in Schlessinger’s famous theorem on the existence of (semi-) universal formal deformations [cf.: M. Schlessinger, Trans. Am. Math. Soc. 130, 208–222 (1968; Zbl 0167.49503)]. This is followed by a discussion of deformation functors and local moduli functors in general, including their obstruction spaces as well as concrete applications to the deformation theory of algebraic surfaces in characteristic zero. In this regard, the author provides a largely self-contained treatment of formal deformation theory along the classical approach, that is without introducing cotangent complexes (à la L. Illusie) or methods of differential graded Lie algebras as more recent tools in deformation theory.

Chapter 3 illustrates the general theory of deformation functors by several concrete examples. With the single exception of deformations of algebraic vector bundles, which have already been exhaustively treated in several recent research monographs like the book of D. Huybrechts and M. Lehn [“The geometry of moduli spaces of sheaves”. Aspects Math. E 31 (1997; Zbl 0872.14002)], the author describes in great detail the most important deformation functors in current algebraic geometry, mainly by carefully verifying Schlessinger’s conditions for the existence of (semi-)universal deformation spaces in the respective cases. Moreover, the first-order deformations, i.e., the tangent spaces of these functors, as well as the corresponding obstruction spaces are thoroughly analyzed, thereby revealing a similar pattern in almost all of these exemplary cases. As for the concrete examples treated in this chapter, the author discusses the deformation functor of an affine scheme with at most quotient singularities, the local (relative) Hilbert functor of closed subschemes of a scheme, the local Picard functor of a scheme, deformations of sections of an invertible sheaf on a scheme, deformations of various types of morphisms of schemes, deformations of a closed embedding, and (co-)stable subschemes.

Chapter 4 gives a thorough introduction to Hilbert schemes, Quot schemes, and flag Hilbert schemes. As the author points out, these objects are needed to construct important examples of global deformations, and to study their local behaviour in the framework of the main theme of the present text. Besides, until now it was rather difficult to give precise references for many results on the geometry of these classifying objects, just like in local deformation theory, and this very fact has been another reason for the author to include that chapter in his book. Together with the very recent monograph “Fundamental Algebraic Geometry: Grothendieck’s FGA Explained” by B. Fantechi et al. [Math. Surv. Monogr. 123 (2005; Zbl 1085.14001)], this chapter in the book under review provides the only (and overdue) reasonably comprehensive exposition on Grothendieck’s Hilbert and Quot schemes. Apart from a general introduction to Castelnuovo-Mumford regularity, flattening stratifications, Hilbert schemes, Quot schemes, flag Hilbert schemes, and Grassmannians, together with applications to families of projective schemes, there is a concluding section on plane curves, their equisingular infinitesimal deformations, and their so-called Severi varieties. The author’s approach to the proof of existence of nodal curves with an arbitrary number of singularities is very original and apparently new. Based on the use of multiple point schemes, this example strikingly illustrates the power of algebraic deformation theory even in classical curve theory.

The five appendices at the end of the book collect some basic standard topics and are titled as follows: A. Flatness; B. Differentials; C. Smoothness; D. Complete intersections; E. Functorial language. Most of the results presented here come with full proofs, which further strengthens the already high degree of self-containedness of the book.

There are no explicit exercises or working problems accompanying the text, but there is a wealth of illustrating concrete examples, additional remarks, historical annotations, and hints to further reading. The bibliography includes 190 references, ranging from the very classical up to the most recent articles and books, and both a very carefully compiled list of used symbols and a just as thorough alphabetical index considerably enhance the value of the entire treatise.

Without any doubt, this is a masterly book on a highly advanced topic in algebraic geometry. The author’s style of writing captivates by its high degree of comprehensiveness, completeness, rigour, sytematical exposition, creative originality, lucidity, and user-friendliness in a like manner. The entire text is kept at a level that makes it suitable for graduate students with a solid background in commutative algebra, homological algebra, and basic algebraic geometry. But even for experts and active researchers in algebraic geometry, this unique book on algebraic deformation theory offers a great deal of inspiration and new insights, too, and its future role as a standard source and reference book in the field can surely be taken for granted from now on.

Reviewer: Werner Kleinert (Berlin)

### MSC:

14-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry |

14D15 | Formal methods and deformations in algebraic geometry |

14B12 | Local deformation theory, Artin approximation, etc. |

14B10 | Infinitesimal methods in algebraic geometry |

14D20 | Algebraic moduli problems, moduli of vector bundles |

14C05 | Parametrization (Chow and Hilbert schemes) |