##
**Sheaves in topology.**
*(English)*
Zbl 1043.14003

Universitext. Berlin: Springer (ISBN 3-540-20665-5/pbk). xvi, 236 p. (2004).

The text under review grew out of a series of lectures given by the author at the University of Bordeaux, France, during the 2000/2001 academic year. The author’s aim is to provide the reader with a systematic introduction to the theory of constructible sheaves in modern algebraic topology and its various applications to singular complex varieties, and that in a comprehensible, concise and user-friendly way. Constructible sheaves, in their abstract categorical setting, appeared in algebraic topology at an early stage, first as local systems of coefficients, and were then developed, in the 1970s, as an important tool in the advanced study of the topology of (singular) complex spaces. Although the vast amount of spectacular applications of abstract sheaves and their cohomology has become sheerly overwhelming, during the last decades, and in spite of the existence of a number of excellent books devoted to this subject, many topologists and geometers seem to be still hesitant to adopt this utmost powerful theory. Admittedly, there is quite a huge formalism behind the modern abstract approach to sheaf theory, ranging from derived categories to perverse sheaves, but the recent developments in algebraic topology, algebraic geometry, complex analytic geometry, micro-local analysis, and mathematical physics can barely be followed up without a profound knowledge of it, on the other hand.

The book under review covers most of the basic notions and results in the theory of constructible sheaves on complex spaces, together with numerous concrete geometric examples and applications. In addition, and along this path, the text does the service of filling the yawning gap between the very complete and highly general monographs [such as the books by M. Kashiwara and P. Shapira, “Sheaves on manifolds” (Berlin 1990; Zbl 0709.18001), A. A. Beilinson, J. Bernstein and P. Deligne, “Faisceaux pervers”, Astérisque 100 (1982; Zbl 0536.14011) or J. Schürmann: “Topology of singular spaces and constructible sheaves” (Basel 2003; Zbl 1041.55001)], a number of recent survey articles on the subject, and some related research papers.

As for the approach chosen in the present introductory treatise, the author tries to take the reader from the simpler, meanwhile classical parts of the theory up to some most recent, powerful and general results, which represent the forefront of research in this realm. In this vein, the reader will meet a number of theorems that appear, step-by-step, in increasing generality and applicability. This methodological strategy is very pleasant, in that it makes the text overall enlightening and instructive, without interrupting its steady flow, but it also forces the author to omit most of the proofs in the first five chapters. As the author points out, this choice of his is not only motivated by the goal to keep the text floating, digestible and concise, but also by the fact that there are enough excellent sources that the reader could (and should) consult for looking up those details. Among those references of nearly encyclopaedic character are the books by M. Kashiwara and P. Shapira and J. Schürmann (loc. cit), which the present text frequently refers to.

However, some results come with full proofs, mainly those that seemed new to the author, and most examples and applications have been provided with full details and ample explanations.

The text consists of six chapters, each of which is divided into several sections.

Chapter 1 gives a brief survey of the theory of derived categories, according to J.-L. Verdier and his successors. Chapter 2 is entitled “Derived categories in topology” and treats generalities on sheaves, derived tensor products, direct and inverse images of sheaves, sheaf cohomology, the adjunction triangle in cohomology, and the basics on local systems (of coefficients) as needed for the construction of perverse sheaves later on.

Chapter 3 discusses the framework of Poincaré-Verdier duality and, in particular, Poincaré and Alexander duality in algebraic topology, together with an exemplification of Borel-Moore homology and a number of cohomological vanishing theorems (after Deligne and Esnault-Viehweg) for later use.

Throughout the text, the author clearly separates the categorical, topological and analytic aspects from each other, thereby helping the reader to grasp the power of the general approach to sheaves.

Chapter 4 turns to the theory of constructible sheaves. Apart from their definition and basic properties, the author explains the triangulated category of bounded constructible complexes on a complex algebraic variety, discusses the functors of nearby and vanishing cycles, relates them to the classical theory of singularities, and he ends this already more advanced chapter with the description of the characteristic variety and the characteristic cycle associated with a constructible sheaf on a smooth manifold. This includes many applications to the topology of manifolds and micro-local analysis.

The theory developed so far culminates in the study of perverse sheaves. This is done in chapter 5, where the reader gets acquainted with the formalism of \(t\)-structures, \(p\)-perverse sheaves, the category of perverse sheaves on a variety, a special case of the Artin vanishing theorem, the fundamentals of the theory of \(D\)-modules and the role of perverse sheaves in there, the Riemann-Hilbert correspondence in a special case, and the basics of intersection (co-)homology after Goresky-MacPherson. In this context, the author also touches upon the corresponding Lefschetz-type theorems and some comparison theorems for intersection cohomology and classical cohomology.

Chapter 6, the concluding part of the book, is the longest and most original chapter of the entire treatise. The author discusses various applications of the general theory of perverse sheaves to concrete geometric situations. This chapter offers old and new results likewise, mainly with a view to hypersurface singularities. Being one of the leading experts in singularity theory, the author (re-)considers the Milnor fibers and the monodromy of isolated singularities, the topology of deformations, the topology of polynomial functions, and the geometry of hyperplane and hypersurface arrangements. This chapter appears very elaborated and detailed, with full proofs and numerous important examples, and it provides many new results, improvements of old results, and comparisons to earlier approaches. Most of the main results here involve properties of constructible or perverse sheaves in an essential way, pointing out the unifying character, elegance, utility, and ubiquity of their theory.

There are also many exercises scattered throughout the text, which are to add useful details to the main text. These exercises mostly come with detailed hints and references for suitable further reading.

No doubt, this is a highly enlightening inviting and useful book. Written in a very lucid, detailed and rigorous style, with an abundance of concrete examples and applications, it serves its afore-mentioned purpose in a perfect way.

Also experts in the field will find quite a bit of novelties in there, especially in the last chapter, and even physicists might profit from studying this largely introductory text.

The book under review covers most of the basic notions and results in the theory of constructible sheaves on complex spaces, together with numerous concrete geometric examples and applications. In addition, and along this path, the text does the service of filling the yawning gap between the very complete and highly general monographs [such as the books by M. Kashiwara and P. Shapira, “Sheaves on manifolds” (Berlin 1990; Zbl 0709.18001), A. A. Beilinson, J. Bernstein and P. Deligne, “Faisceaux pervers”, Astérisque 100 (1982; Zbl 0536.14011) or J. Schürmann: “Topology of singular spaces and constructible sheaves” (Basel 2003; Zbl 1041.55001)], a number of recent survey articles on the subject, and some related research papers.

As for the approach chosen in the present introductory treatise, the author tries to take the reader from the simpler, meanwhile classical parts of the theory up to some most recent, powerful and general results, which represent the forefront of research in this realm. In this vein, the reader will meet a number of theorems that appear, step-by-step, in increasing generality and applicability. This methodological strategy is very pleasant, in that it makes the text overall enlightening and instructive, without interrupting its steady flow, but it also forces the author to omit most of the proofs in the first five chapters. As the author points out, this choice of his is not only motivated by the goal to keep the text floating, digestible and concise, but also by the fact that there are enough excellent sources that the reader could (and should) consult for looking up those details. Among those references of nearly encyclopaedic character are the books by M. Kashiwara and P. Shapira and J. Schürmann (loc. cit), which the present text frequently refers to.

However, some results come with full proofs, mainly those that seemed new to the author, and most examples and applications have been provided with full details and ample explanations.

The text consists of six chapters, each of which is divided into several sections.

Chapter 1 gives a brief survey of the theory of derived categories, according to J.-L. Verdier and his successors. Chapter 2 is entitled “Derived categories in topology” and treats generalities on sheaves, derived tensor products, direct and inverse images of sheaves, sheaf cohomology, the adjunction triangle in cohomology, and the basics on local systems (of coefficients) as needed for the construction of perverse sheaves later on.

Chapter 3 discusses the framework of Poincaré-Verdier duality and, in particular, Poincaré and Alexander duality in algebraic topology, together with an exemplification of Borel-Moore homology and a number of cohomological vanishing theorems (after Deligne and Esnault-Viehweg) for later use.

Throughout the text, the author clearly separates the categorical, topological and analytic aspects from each other, thereby helping the reader to grasp the power of the general approach to sheaves.

Chapter 4 turns to the theory of constructible sheaves. Apart from their definition and basic properties, the author explains the triangulated category of bounded constructible complexes on a complex algebraic variety, discusses the functors of nearby and vanishing cycles, relates them to the classical theory of singularities, and he ends this already more advanced chapter with the description of the characteristic variety and the characteristic cycle associated with a constructible sheaf on a smooth manifold. This includes many applications to the topology of manifolds and micro-local analysis.

The theory developed so far culminates in the study of perverse sheaves. This is done in chapter 5, where the reader gets acquainted with the formalism of \(t\)-structures, \(p\)-perverse sheaves, the category of perverse sheaves on a variety, a special case of the Artin vanishing theorem, the fundamentals of the theory of \(D\)-modules and the role of perverse sheaves in there, the Riemann-Hilbert correspondence in a special case, and the basics of intersection (co-)homology after Goresky-MacPherson. In this context, the author also touches upon the corresponding Lefschetz-type theorems and some comparison theorems for intersection cohomology and classical cohomology.

Chapter 6, the concluding part of the book, is the longest and most original chapter of the entire treatise. The author discusses various applications of the general theory of perverse sheaves to concrete geometric situations. This chapter offers old and new results likewise, mainly with a view to hypersurface singularities. Being one of the leading experts in singularity theory, the author (re-)considers the Milnor fibers and the monodromy of isolated singularities, the topology of deformations, the topology of polynomial functions, and the geometry of hyperplane and hypersurface arrangements. This chapter appears very elaborated and detailed, with full proofs and numerous important examples, and it provides many new results, improvements of old results, and comparisons to earlier approaches. Most of the main results here involve properties of constructible or perverse sheaves in an essential way, pointing out the unifying character, elegance, utility, and ubiquity of their theory.

There are also many exercises scattered throughout the text, which are to add useful details to the main text. These exercises mostly come with detailed hints and references for suitable further reading.

No doubt, this is a highly enlightening inviting and useful book. Written in a very lucid, detailed and rigorous style, with an abundance of concrete examples and applications, it serves its afore-mentioned purpose in a perfect way.

Also experts in the field will find quite a bit of novelties in there, especially in the last chapter, and even physicists might profit from studying this largely introductory text.

Reviewer: Werner Kleinert (Berlin)

### MSC:

14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |

14F43 | Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) |

14F17 | Vanishing theorems in algebraic geometry |

14B05 | Singularities in algebraic geometry |

32C18 | Topology of analytic spaces |

14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |

14F10 | Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials |

55R65 | Generalizations of fiber spaces and bundles in algebraic topology |