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Residues and $$\mathcal D$$-modules. (English) Zbl 1069.32001
Laudal, Olav Arnfinn (ed.) et al., The legacy of Niels Henrik Abel. Papers from the Abel bicentennial conference, University of Oslo, Oslo, Norway, June 3–8, 2002. Berlin: Springer (ISBN 3-540-43826-2/hbk). 605-651 (2004).
Author’s abstract: The work ‘Mémoire sur une propriété generale d’une classe très étendue de fonctions transcendantes’ by Niels Henrik Abel has started a new era in which geometry, algebra and complex analysis are brought together. It is remarkable that Abel already in 1826 described the process of integrating algebraic functions over cycles – and more generally over chains – which eliminate some variables while the function of the remaining variables has a specific transcendental nature. For example, the conclusive result from 1965 by N. Nilsson in [Ark. Mat. 5, 463–476 (1965; Zbl 0168.42004)] was already suggested in Abel’s work.
Multidimensional residue theory is a subject which also has developed in the spirit of Abel. This article describes how regular holonomic $${\mathcal D}_X$$-modules can be realised by distributions satisfying regular holonomic systems on a complex manifold $$X$$. Such distributions emerge from Nilsson class functions, i.e., multi-valued analytic functions defined in the complement of a hypersurface $$T$$ with finite determination and moderate growth along $$T$$.
For the entire collection see [Zbl 1047.00019].

##### MSC:
 32C38 Sheaves of differential operators and their modules, $$D$$-modules 32A27 Residues for several complex variables 32-03 History of several complex variables and analytic spaces 01A60 History of mathematics in the 20th century 14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials