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Local Systems and Constructible Sheaves Local Systems and Constructible Sheaves Fouad El Zein and Jawad Snoussi Abstract. The article describes Local Systems, Integrable Connections, the equivalence of both categories and their relations to Linear di®erential equa- tions. We report in details on Regular Singularities of Connections and on Singularities of local systems which leads to the theory of Intermediate ex- tensions and the Decomposition theorem. Mathematics Subject Classi¯cation (2000). Primary 32S60,32S40, Secondary 14F40. Keywords. Algebraic geometry, Analytic geometry, Local systems, Linear Dif- ferential Equations, Connections, constructible sheaves, perverse sheaves, Hard Lefschetz theorem. Introduction The purpose of these notes is to indicate a path for students which starts from a basic theory in undergraduate studies, namely the structure of solutions of Linear Di®erential Equations which is a classical subject in mathematics (see Ince [9]) that has been constantly enriched with developments of various theories and ends in a subject of research in contemporary mathematics, namely perverse sheaves. We report in these notes on the developments that occurred with the introduction of sheaf theory and vector bundles in the works of Deligne [4] and Malgrange [3,2)]. Instead of continuing with di®erential modules developed by Kashiwara and ex- plained in [12], a subject already studied in a Cimpa school, we shift our attention to the geometrical aspect represented by the notion of Local Systems which de- scribe on one side the structure of solutions of linear di®erential equations and on the other side the cohomological higher direct image of a constant sheaf by a proper smooth di®erentiable morphism. Then we introduce the theory of Connections on vector bundles generalizing to analytic varieties the theory of linear di®erential equations on a complex disc. The de¯nition of local systems is easily extended to varieties of dimension n while it is more elaborate to extend the notion of di®erential equations into the concept 2 Fouad El Zein and Jawad Snoussi of connections. Deligne establish an equivalence of categories between local sys- tems and integrable connections. In particular this point of view explains how the connections, named after Gauss and Manin, are de¯ned by the cohomology of families of algebraic or analytic va- rieties ( precisely by a smooth proper morphism). A background to this result is the classical construction of solutions of di®erential equations as integrals along cycles of relative di®erential forms on algebraic families of varieties de¯ned by a smooth proper morphism. DeRham resolutions of local systems are obtained via the associated integrable connections. Singularities in the ¯elds of algebraic and analytic geometry appear in the study of linear di®erential equations with meromorphic coe±cients on the punctured complex disc. In particular a basic result of Fuchs on equations with Regular Singularity is at the origin of the theory and leads to the notion of meromorphic connections with regular singularity. The work of P. Deligne in 1970 [4] pointed out to the developments of this theory to higher dimensional varieties in algebraic and analytic geometry. Constructible sheaves. Singularity theory in mathematics which arise for example with the vanishing of the di®erential of a morphism, has had important develop- ments in algebraic geometry; in particular Whitney's and Thom's strati¯cation theory [10] contributed to a further generalization of local systems, namely the concept of constructible sheaves which appear in the study of cohomology the- ory of the ¯bers of any algebraic morphism. This concept is used in s¶eminairede g¶eom¶etriealg¶ebrique[6] by Grothendieck's school and in an important article [11] on Chern classes for singular algebraic varieties by MacPherson. Among the complexes of sheaves with constructible cohomology, the perverse sheaves have important special properties since they are related to the theory of di®erential modules in the sense that the DeRham complex de¯ned by an holo- nomic di®erential module is a complex with constructible cohomology sheaves which is in fact a perverse sheaf. Complexes with constructible cohomology sheaves are preserved by derived direct image by a proper algebraic morphism (and in general by the six classical oper- ations). The concept and the proofs are based on Thom-whitney strati¯cation of varieties and morphisms and a result proved by Mather known as Thom - Mather isotopy lemma describing local topological triviality along strata. Decomposition theorem. This theorem is stated here to illustrate how it is possible to develop a basic classical result such as Lefschetz theorem via the above tools. The proof is beyond the scope of this exposition. The reader don't see here the use of regularity necessary in the proof, neither we can present Hodge theory which is hidden in the hypothesis of geometric local system. In fact we mention further references where it is possible to ¯nd more results on the subject. Family of Elliptic Curves. The appendix gives explicit computation of the mon- odromy of the local system and the Gauss-Manin connection de¯ned by the family Local Systems and Constructible Sheaves 3 of elliptic curves, a mathematical subject that should serve as a test example for every mathematician. Contents. 1) Local Systems. 1.1. Background in undergraduate studies 1.2. De¯nition and properties 1.3. Local systems and Representations of the fundamental group 1.4. System of n¡linear ¯rst order di®erential equations and Local Systems 1.5. Connections and Local Systems 1.6. Fibrations and Local Systems (Gauss-Manin Connection) 2) Singularities of local systems and Systems of di®erential equations with mero- morphic coe±cients: Regularity 2.1. Systems with meromorphic coe±cients on the complex disc 2.2. Connections with Logarithmic singularities 2.3. Meromorphic connections on the disc 2.4. Regular meromorphic connections. 3)Singularities of local systems: Constructible Sheaves. 3.1. Strati¯cation theory 3.2. Cohomologically Constructible sheaves. 4) Decomposition theorem. Appendix. Example: Family of Elliptic Curves. The reader can expect to learn from this expository paper various points of view of the subject in topology, geometry and analysis in the direction of the decompo- sition theorem. To cover recent developments in the theory, the reader dispose of various books listed after the references. Finally, the theory of Local Systems and Constructible Sheaves play an impor- tant role in the theory of Arrangement of Hyperplanes and we refer the reader to expository and research articles by experts on this subject in this summer school. 1. Local Systems We study here sheaves of groups with topological interest known as local sys- tems or locally constant sheaves. They arise in mathematics as solutions of linear di®erential equations, as higher direct image of a constant sheaf by a proper dif- ferentiable submersive morphism of manifolds and as representations of the funda- mental group of a topological space. Local systems can be enriched with structures reflecting geometry like the notion of Hodge structures. 1.1. Background in undergraduate studies The a±ne di®erential equation zu0(z) = 1 with complex variable z, well known by students, is singular at the origin, since we can apply Cauchy's theorem on 4 Fouad El Zein and Jawad Snoussi the exitence of a unique solution with given initial condition only for z 6= 0. We 0 1 put u (z) = z , then for any a 6= 0 there exists an analytic solution near a for n P (¡1) n+1 jz ¡ aj < jaj, u(z) = n¸0 (n+1)an+1 (z ¡ a) . In particular, for a = 1 we de¯ne in this way the function u(z) = logz solution of the equation satisfying the condition u(1) = 0. Then we can extend the above local solution into the global function log z = r + i; 2] ¡ ¼; ¼[ for z = reiθ. The main point of interest in our study, due to the singularity of the equation at zero, comes down in this case to the fact that logz cannot be extended in a continuous function beyond the above domain in the complex plane, since its limit near a negative real number ¡r along a path in the upper half plane is log r+i¼ and di®ers by 2i¼ with its limit log r¡i¼ along a path in the opposite half plane. Such function is an inverse to the exponential map ez, but other inverse maps can be written as logz + 2ki¼ and are always de¯ned on C¡ f ray g. They are called various determinations of the logarithm. The exponential map ez : C ! C¤ is said to be a covering and a determination of log z is a section of such covering. However our interest is in linear di®erential equations, for example zu0(z)¡®u(z) = 0, for ® 2 R, with solutions z® = r®eiαθ. When we cross the negative reals the solution is multiplied by ei2¼®. We express this property by introducing the one dimensional vector space Cz® of all the solutions de¯ned on a simply connected open subset of C¤ and the linear endomorphism T : Cz® ! Cz® called monodromy, acting as ei2¼®Id on this linear space. In another point of view, the monodromy extends to a morphism from Z to the group of linear automorphisms of the one dimensional vector space Cz® de¯ned by n 7! T n. We obtain in this special case the representation of the fundamental ¤ group ¼1(C ) identi¯ed with Z, de¯ned by the di®erential equation. 1.2. De¯nition and properties To de¯ne local systems we use the language of sheaf theory for which basic refer- ences are Godement [5] and Warner [14], then we describe here the relation with the topology of the base space, precisely the fundamental group.
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