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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 .
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.
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.
KeywordsExact Sequence Local System Good Covering Global Section Moderate Growth
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