# Deligne modules

• Jan-Erik Björk
Chapter
Part of the Mathematics and Its Applications book series (MAIA, volume 247)

## Summary

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

In Section 1 we define a subheaf whose sections have moderate growth along T. This gives a sheaf denoted by
and called the Deligne extension of the connection 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.

In addition to the moderate growth condition we consider sections of which are locally 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\TL). 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
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Notes

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Deligne, P., Théorèmes de Lefschetz et critères de degénerescence de suites spectrales, Publ Math IHES 35, 1968.Google Scholar
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