Analytic *D*-Modules and Applications
pp 167-192 |
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# Deligne modules

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## Summary

In this chapter we construct a family of holonomic *D* _{ X }-modules associated to pairs (*T*, *L*), where *T* ⊂ *X* is an analytic hypersurface and *L* a local system in *X*\*T*. Given such a pair there exists the direct image sheaf .

*moderate growth*along

*T*. This gives a sheaf denoted by

*O*

_{ X\T }⨂

*L*. We prove that every Deligne extension is a left

*D*

_{ X }-module. The case when the hypersurface

*T*has

*normal crossings*is studied in section 2 and 3. Several results about the left

*D*

_{ X }-module structure on Deligne sheaves are established. In particular we prove that every Deligne sheaf is a holonomic

*D*

_{ X }-module when

*T*has normal crossings.

*Desingularisation* is used in Section 4 to extend results in the normal crossing case to arbitrary analytic hypersurfaces. In this way we obtain an extensive class of holonomic *D* _{ X }-modules. A number of results concerned with the holonomic dual and other submodules occur. Of particular importance are the *minimal Deligne extensions*. For a given pair (*T*, *L*) there exists a unique largest holonomic *D* _{ X }-submodule of Del (*O* _{ X\T }⨂*L*) whose holonomic dual has no torsion. This submodule is denoted by. *M* _{⨂}(*T*,*L*) and is called the *minimal Deligne extension* of the connection *O* _{ X\T }⨂*L*).

In section 4 we also prove a *Hartog’s Theorem* which asserts that a section of satisfying the local moderate growth condition at a dense open subset of the regular part of *T* is a section of the Deligne module. We also discuss the interplay between Deligne modules and *Nilsson class functions* which consist of multi-valued analytic functions with finite determination satisfying the moderate growth condition along their polar sets.

*square-integrable*when they are expressed in trivialisations of the local system. This gives a sheaf denoted by

*L*

^{2}(

*T*,

*L*) and we prove that it is a coherent

*O*

_{X}-submodule of Del(

*O*

_{X\T}⨂

*L*). An important result is the equality:

The existence and the properties of Deligne modules will be used in Chapter V to study regular holonomic modules, where Deligne modules is a generating class when direct images are included to generate *D* _{ X }-modules supported by analytic sets of positive codimension.

## Keywords

Exact Sequence Local System Good Covering Global Section Moderate Growth## Preview

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## Notes

- [4]Pham, F., Singularités des systemes différentiels de Gauss-Manin, Birkhäuser, 1979.Google Scholar
- [8]Mebkhout, Z., La formalisme des six operations de Grothendieck pour les D-modules cohérents, Hermann, Paris, 1989.Google Scholar
- Poly, J. B., Sur l’homologie des courants a support dans un ensemble semi-analytique, Bull. Soc. Math. France Memoire
**38**(1974), 35–43.MathSciNetzbMATHGoogle Scholar - [2]Deligne, P., Théorèmes de Lefschetz et critères de degénerescence de suites spectrales, Publ Math IHES
**35**, 1968.Google Scholar - [1]Nilsson, N., Some growth and ramification properties of certain integrals on algebraic manifolds, Arkiv für Mat.
**5**(1965), 463–476.MathSciNetzbMATHCrossRefGoogle Scholar - [2]Nilsson, N., Monodromy and asymptotic integrals, ibid.
**18**(1980), 181–198.MathSciNetzbMATHGoogle Scholar