##
**Sheaves on manifolds. With a short history “Les débuts de la théorie des faisceaux” by Christian Houzel.**
*(English)*
Zbl 0709.18001

Grundlehren der Mathematischen Wissenschaften, 292. Berlin etc.: Springer-Verlag. x, 512 p. DM 168.00 (1990).

This treatise, modestly entitled “Sheaves on manifolds” is dedicated, in fact, to the study of sheaves from the microlocal point of view, and certainly will soon become a classic. Even if a lot of notions and results are “classical” (many of them appearing in the fundamental “S- K-K” paper) this is the first place where they are coherently and systematically exposed.

A lot of results presented here are due to the two authors, scattered in different papers. But briefly about the content of the book. It begins with an interesting historical note (in French) written by Christian Houzel: “Les débuts de la théorie des faisceaux”, which goes up to the notion of derived category.

Chapter I contains the homological algebra necessary in the sequel. On 82 pages (of which 12 are devoted to exercises, in fact results which complete the theory), the authors introduce the notions of derived category, derived functor, etc. A paragraph is devoted to the Mittag- Leffer condition, fundamental in the sequel.

Chapter II, “Sheaves”, develops, after the usual definitions, the theory in the language of derived categories, including the “six operations” (in the terminology of Grothendieck). Besides the carefully explained classical things, we remark the non-characteristic deformation lemma, as well as other technical results which will be systematically used in the next chapters. Čech cohomology is introduced briefly (though it will no more be used), and a last paragraph gives some classical and important examples of sheaves in real and complex manifolds.

Chapter III is dedicated to Poincaré-Verdier duality and Fourier-Sato transform. The authors, following Verdier, construct a right adjoint \(f^ !\) to the functor \({\mathbb{R}}F_ !\), where f is a continuous map f: \(Y\to X\) of locally compact spaces, verifying some suitable conditions. This is the Poincaré-Verdier duality. This duality is carefully studied; the authors introduce the “dualizing complex” on Y, and they show that it is isomorphic to the orientation sheaf of Y, shifted by the dimension. In § 3.5 the \(\gamma\)-topology is introduced, and in § 3.6 some equivalences of categories are established using “kernels” (see the example 3.6.6). The Fourier-Sato transform, which interchanges (conic) sheaves on vector bundles and sheaves on the dual vector bundle, is studied very carefully, and this is the first place where this topic is systematically explained.

The next chapter, “Specialization and microlocalization”, introduces Sato’s functors of specialization \(\nu_ M(F)\) (along a manifold M) and microlocalization \(\mu_ M(F)\) (which is the Fourier-Sato transform of \(\nu_ M(F))\) of a sheaf F (in fact \(F\in Ob(D^ b(X))\). The sheaf \(\nu_ M(F)\) lives on \(T_ MX\). In fact everything is considered in the corresponding derived categories. The basic geometric construction is (for a closed submanifold M of X, X being also a manifold) that of \(\tilde X_ M\), the normal deformation of M in X. After that the authors study the functor \(\mu\) hom introduced by them in the paper “Microlocal study of sheaves” [Astérisque 128 (1985; Zbl 0589.32019)], which generalizes \(\nu_ M\) and will be fundamental in the sequel. We do not give here the precise definition of this functor, observing that the stalk of \(\mu\) hom(F,G) can be described using \(\gamma\)-topologies.

In Chapter V, the authors study the micro-support of sheaves (introduced by them in the same Astérisque paper). For a sheaf F, its micro-support SS(F) lives on \(T^*X\) and, roughly speaking, it describes the set of codirections in which F “does not propagate”. There are 3 equivalent definitions of SS(F), the non-characteristic deformation lemma being used to prove these equivalences. Using the same lemma and the \(\gamma\)- topology, in the case of open subsets of a finite dimensional vector space the authors derive a microlocal extension theorem, a microlocal cut-off lemma for sheaves etc. After a thorough study of the functorial properties of the micro-support (in the non-characteristic case for inverse images, and in the proper case for direct images), a beautiful application, the Morse inequality for sheaves (Prop. 5.4.20), is given.

Chapter VI studies the relations between the micro-support and microlocalization. The main result here is the involutivity of SS(F) in \(T^*X\); before proving that, one finds the use of micro-supports for localizing \(D^ b(X)\) with respect to an open set \(\Omega\) of the cotangent bundle which allows to define “microlocal inverses” or direct images. The functorial behavior of the micro-support is studied removing the non-characteristic restriction. One shows that supp(\(\mu\) hom(F,F))\(\subset SS(F)\), and that the microsupport \(\mu_ M(F)\) is contained in the normal cone of SS(F) along \(T^*_ MX\). This last result is the sheaf-version of a theorem on micro-hyperbolic systems [the authors, Acta Mat. 142, 1-55 (1979; Zbl 0413.35049)].

The 7th chapter, “Contact transformation and pure sheaves”, shows that if \(\Omega_ X\), \(\Omega_ Y\) are open subsets of \(T^*X\), respectively \(T^*Y\), and \(\chi\) is a contact transformation between them, under suitable conditions one can construct an isomorphism between \(D^ b(X,\Omega_ X)\) and \(D^ b(Y,\Omega_ Y)\), and this isomorphism is compatible with \(\mu\) hom. For doing this, the authors first microlize the construction of kernels done in Chapter II, and then study the microlocal composition of kernels. The image of the constant sheaf on a submanifold M of X leads naturally to the notion of pure sheaf along a smooth Lagrangian manifold. Generically, a pure sheaf is isomorphic to some \(L_ M[d]\), for some A-module L and some shift d, but determining this d involves the whole machinery of the inertia index of a triplet of Lagrangian planes. The theory of this chapter (which, in a different form, was introduced in the Astérisque paper [loc.cit.]) may be considered in the sheaf-theoretic version of the theory of Fourier integral operators, or of simple holonomic modules of “SSK”.

The last 3 chapters are dedicated to (very interesting) applications of the theory already developed.

Chapter VIII, “Constructible sheaves”, contains a very detailed study of these objects on real manifolds. The notion of \(\mu\)-stratification is introduced. This is a modification of the concept of a Whitney stratification, in the sense that one asks a suitable condition on the conormal bundles of the strates to be fulfilled. A paragraph is dedicated to subanalytic sets, and another, very interesting, to subanalytic isotropic sets and \(\mu\)-stratification. One main theorem is on the existence, for every locally finite covering of the manifold X by subanalytic sets, of a finer \(\mu\)-stratification. After these geometric preliminaries the authors study \({\mathbb{R}}\)-constructible sheaves, \({\mathbb{C}}\)-constructible sheaves, and the relation between the functor of duality and \(\mu\) hom for constructible sheaves. Another main result is on the equivalence between the facts that SS(F) is a conic closed \({\mathbb{C}}\)-analytic Lagrangian subset, that SS(F) is a closed \(C^*\)- conic subanalytic \({\mathbb{R}}\)-isotropic subset, that SS(F) is conic and \(F\in ob(D^ b_{w-{\mathbb{P}}c}(X))\), or finally that there exists a stratification of X by \({\mathbb{C}}\)-analytic subsets \(X_ j\) such that \(H^ k(F)|_{X_ j}\) are locally constant. The last paragraph of this chapter considers the nearby-cycle functor and the vanishing-cycle functor. One interesting point is that one recovers SS(F) from the vanishing-cycle functor.

Chapter IX introduces, on a real analytic manifold X, the notion of subanalytic chains, with the help of the dualizing couplex \(\omega_ X\). Let F be an \({\mathbb{R}}\)-constructible object in \(D^ b(X)\), and CC(F) the characteristic cycle of F (which lives in \(H^ 0_{SS(F)}(T^*X,\pi^{-1}\omega_ X)\) as the image of \(id_ F\) of Hom(F,F) in \(H^ 0_{SS(F)}\). Then, after studying the functorial properties of CC(F), the authors show that if M is a closed submanifold of X and \(V\in Ob(D^ b(Mod^ f(k))\)- when the base ring A is a field K of characteristic zero - then \(CC(V_ M)=m[T^*_ MX]\) where \(T^*_ MX\) is the Lagrangian cycle associated to \(T^*_ MX\), and m is just the Euler-Poincaré characteristic \(\sum_{j}(-1)^ j\dim H^ j(V).\) In general, this \(\chi\) (X;F) can be obtained as the intersection number of CC(F) and the cycle associated to the zero section of \(T^*X\). A version of the Lefschetz fixed point formula is given in this context. As a particular case (Example 9.6.6) one obtains the classical Lefschetz fixed point formula. The results of this paragraph, announced by the first author [Adv. Stud. Pure Math. 14, 369-378 (1988; Zbl 0699.22024)] are presented here for the first time.

Chapter X deals with perverse sheaves. The authors introduce also a kind of microlocal perversity. The things are too technical to be explained here. Let us say that everybody interested in the subject should carefully examine this chapter.

The last chapter (Chapter XI) contains applications to O- and \({\mathcal D}\)- modules. We should mention here two results: The propagation of singularities theorem for a coherent \({\mathcal D}_ X\)-module (X being a complex manifold (Th. 11.3.3)) and the perversity of the couplex of holomorphic solutions of a holonomic module \({\mathfrak M}\) (Th. 11.3.7). One finds also a microlocal study of \(O_ X\), a short paragraph on microfunctions, etc.

The treatise ends with a substantial appendix, which deals with symplectic geometry.

Every chapter ends with a set of interesting exercises (even the appendix!) and also with a historical note.

The book, which contains a lot of new results (of which I have pointed a few) show how results which initially stemmed from the study of differential operators (so results from analysis) have in fact a much more general and purely geometric meaning. Thus, once more and in a central field of mathematics, the underlying unity of mathematics is revealed.

A lot of results presented here are due to the two authors, scattered in different papers. But briefly about the content of the book. It begins with an interesting historical note (in French) written by Christian Houzel: “Les débuts de la théorie des faisceaux”, which goes up to the notion of derived category.

Chapter I contains the homological algebra necessary in the sequel. On 82 pages (of which 12 are devoted to exercises, in fact results which complete the theory), the authors introduce the notions of derived category, derived functor, etc. A paragraph is devoted to the Mittag- Leffer condition, fundamental in the sequel.

Chapter II, “Sheaves”, develops, after the usual definitions, the theory in the language of derived categories, including the “six operations” (in the terminology of Grothendieck). Besides the carefully explained classical things, we remark the non-characteristic deformation lemma, as well as other technical results which will be systematically used in the next chapters. Čech cohomology is introduced briefly (though it will no more be used), and a last paragraph gives some classical and important examples of sheaves in real and complex manifolds.

Chapter III is dedicated to Poincaré-Verdier duality and Fourier-Sato transform. The authors, following Verdier, construct a right adjoint \(f^ !\) to the functor \({\mathbb{R}}F_ !\), where f is a continuous map f: \(Y\to X\) of locally compact spaces, verifying some suitable conditions. This is the Poincaré-Verdier duality. This duality is carefully studied; the authors introduce the “dualizing complex” on Y, and they show that it is isomorphic to the orientation sheaf of Y, shifted by the dimension. In § 3.5 the \(\gamma\)-topology is introduced, and in § 3.6 some equivalences of categories are established using “kernels” (see the example 3.6.6). The Fourier-Sato transform, which interchanges (conic) sheaves on vector bundles and sheaves on the dual vector bundle, is studied very carefully, and this is the first place where this topic is systematically explained.

The next chapter, “Specialization and microlocalization”, introduces Sato’s functors of specialization \(\nu_ M(F)\) (along a manifold M) and microlocalization \(\mu_ M(F)\) (which is the Fourier-Sato transform of \(\nu_ M(F))\) of a sheaf F (in fact \(F\in Ob(D^ b(X))\). The sheaf \(\nu_ M(F)\) lives on \(T_ MX\). In fact everything is considered in the corresponding derived categories. The basic geometric construction is (for a closed submanifold M of X, X being also a manifold) that of \(\tilde X_ M\), the normal deformation of M in X. After that the authors study the functor \(\mu\) hom introduced by them in the paper “Microlocal study of sheaves” [Astérisque 128 (1985; Zbl 0589.32019)], which generalizes \(\nu_ M\) and will be fundamental in the sequel. We do not give here the precise definition of this functor, observing that the stalk of \(\mu\) hom(F,G) can be described using \(\gamma\)-topologies.

In Chapter V, the authors study the micro-support of sheaves (introduced by them in the same Astérisque paper). For a sheaf F, its micro-support SS(F) lives on \(T^*X\) and, roughly speaking, it describes the set of codirections in which F “does not propagate”. There are 3 equivalent definitions of SS(F), the non-characteristic deformation lemma being used to prove these equivalences. Using the same lemma and the \(\gamma\)- topology, in the case of open subsets of a finite dimensional vector space the authors derive a microlocal extension theorem, a microlocal cut-off lemma for sheaves etc. After a thorough study of the functorial properties of the micro-support (in the non-characteristic case for inverse images, and in the proper case for direct images), a beautiful application, the Morse inequality for sheaves (Prop. 5.4.20), is given.

Chapter VI studies the relations between the micro-support and microlocalization. The main result here is the involutivity of SS(F) in \(T^*X\); before proving that, one finds the use of micro-supports for localizing \(D^ b(X)\) with respect to an open set \(\Omega\) of the cotangent bundle which allows to define “microlocal inverses” or direct images. The functorial behavior of the micro-support is studied removing the non-characteristic restriction. One shows that supp(\(\mu\) hom(F,F))\(\subset SS(F)\), and that the microsupport \(\mu_ M(F)\) is contained in the normal cone of SS(F) along \(T^*_ MX\). This last result is the sheaf-version of a theorem on micro-hyperbolic systems [the authors, Acta Mat. 142, 1-55 (1979; Zbl 0413.35049)].

The 7th chapter, “Contact transformation and pure sheaves”, shows that if \(\Omega_ X\), \(\Omega_ Y\) are open subsets of \(T^*X\), respectively \(T^*Y\), and \(\chi\) is a contact transformation between them, under suitable conditions one can construct an isomorphism between \(D^ b(X,\Omega_ X)\) and \(D^ b(Y,\Omega_ Y)\), and this isomorphism is compatible with \(\mu\) hom. For doing this, the authors first microlize the construction of kernels done in Chapter II, and then study the microlocal composition of kernels. The image of the constant sheaf on a submanifold M of X leads naturally to the notion of pure sheaf along a smooth Lagrangian manifold. Generically, a pure sheaf is isomorphic to some \(L_ M[d]\), for some A-module L and some shift d, but determining this d involves the whole machinery of the inertia index of a triplet of Lagrangian planes. The theory of this chapter (which, in a different form, was introduced in the Astérisque paper [loc.cit.]) may be considered in the sheaf-theoretic version of the theory of Fourier integral operators, or of simple holonomic modules of “SSK”.

The last 3 chapters are dedicated to (very interesting) applications of the theory already developed.

Chapter VIII, “Constructible sheaves”, contains a very detailed study of these objects on real manifolds. The notion of \(\mu\)-stratification is introduced. This is a modification of the concept of a Whitney stratification, in the sense that one asks a suitable condition on the conormal bundles of the strates to be fulfilled. A paragraph is dedicated to subanalytic sets, and another, very interesting, to subanalytic isotropic sets and \(\mu\)-stratification. One main theorem is on the existence, for every locally finite covering of the manifold X by subanalytic sets, of a finer \(\mu\)-stratification. After these geometric preliminaries the authors study \({\mathbb{R}}\)-constructible sheaves, \({\mathbb{C}}\)-constructible sheaves, and the relation between the functor of duality and \(\mu\) hom for constructible sheaves. Another main result is on the equivalence between the facts that SS(F) is a conic closed \({\mathbb{C}}\)-analytic Lagrangian subset, that SS(F) is a closed \(C^*\)- conic subanalytic \({\mathbb{R}}\)-isotropic subset, that SS(F) is conic and \(F\in ob(D^ b_{w-{\mathbb{P}}c}(X))\), or finally that there exists a stratification of X by \({\mathbb{C}}\)-analytic subsets \(X_ j\) such that \(H^ k(F)|_{X_ j}\) are locally constant. The last paragraph of this chapter considers the nearby-cycle functor and the vanishing-cycle functor. One interesting point is that one recovers SS(F) from the vanishing-cycle functor.

Chapter IX introduces, on a real analytic manifold X, the notion of subanalytic chains, with the help of the dualizing couplex \(\omega_ X\). Let F be an \({\mathbb{R}}\)-constructible object in \(D^ b(X)\), and CC(F) the characteristic cycle of F (which lives in \(H^ 0_{SS(F)}(T^*X,\pi^{-1}\omega_ X)\) as the image of \(id_ F\) of Hom(F,F) in \(H^ 0_{SS(F)}\). Then, after studying the functorial properties of CC(F), the authors show that if M is a closed submanifold of X and \(V\in Ob(D^ b(Mod^ f(k))\)- when the base ring A is a field K of characteristic zero - then \(CC(V_ M)=m[T^*_ MX]\) where \(T^*_ MX\) is the Lagrangian cycle associated to \(T^*_ MX\), and m is just the Euler-Poincaré characteristic \(\sum_{j}(-1)^ j\dim H^ j(V).\) In general, this \(\chi\) (X;F) can be obtained as the intersection number of CC(F) and the cycle associated to the zero section of \(T^*X\). A version of the Lefschetz fixed point formula is given in this context. As a particular case (Example 9.6.6) one obtains the classical Lefschetz fixed point formula. The results of this paragraph, announced by the first author [Adv. Stud. Pure Math. 14, 369-378 (1988; Zbl 0699.22024)] are presented here for the first time.

Chapter X deals with perverse sheaves. The authors introduce also a kind of microlocal perversity. The things are too technical to be explained here. Let us say that everybody interested in the subject should carefully examine this chapter.

The last chapter (Chapter XI) contains applications to O- and \({\mathcal D}\)- modules. We should mention here two results: The propagation of singularities theorem for a coherent \({\mathcal D}_ X\)-module (X being a complex manifold (Th. 11.3.3)) and the perversity of the couplex of holomorphic solutions of a holonomic module \({\mathfrak M}\) (Th. 11.3.7). One finds also a microlocal study of \(O_ X\), a short paragraph on microfunctions, etc.

The treatise ends with a substantial appendix, which deals with symplectic geometry.

Every chapter ends with a set of interesting exercises (even the appendix!) and also with a historical note.

The book, which contains a lot of new results (of which I have pointed a few) show how results which initially stemmed from the study of differential operators (so results from analysis) have in fact a much more general and purely geometric meaning. Thus, once more and in a central field of mathematics, the underlying unity of mathematics is revealed.

Reviewer: G.Gussi

### MSC:

18-02 | Research exposition (monographs, survey articles) pertaining to category theory |

18F20 | Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) |

55N30 | Sheaf cohomology in algebraic topology |

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

58J40 | Pseudodifferential and Fourier integral operators on manifolds |

46F15 | Hyperfunctions, analytic functionals |

35S10 | Initial value problems for PDEs with pseudodifferential operators |

32A45 | Hyperfunctions |