Deligne modules

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


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.


Exact Sequence Local System Good Covering Global Section Moderate Growth 
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Copyright information

© Springer Science+Business Media Dordrecht 1993

Authors and Affiliations

  • Jan-Erik Björk
    • 1
  1. 1.Department of MathematicsStockholm UniversityStockholmSweden

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