Residues of Logarithmic Differential Forms in Complex Analysis and Geometry

Residues of Logarithmic Differential Forms in Complex Analysis and Geometry

Year:    2014

Analysis in Theory and Applications, Vol. 30 (2014), Iss. 1 : pp. 34–50

Abstract

In the article, we discuss basic concepts of the residue theory of logarithmic and multi-logarithmic differential forms, and describe some aspects of the theory, developed by the author in the past few years. In particular, we introduce the notion of logarithmic differential forms with the use of the classical de Rham lemma and give an explicit description of regular meromorphic differential forms in terms of residues of logarithmic or multi-logarithmic differential forms with respect to hypersurfaces, complete intersections or pure-dimensional Cohen-Macaulay spaces. Among other things, several useful applications are considered, which are related with the theory of holonomic $\mathscr{D}$-modules, the theory of Hodge structures, the theory of residual currents and others.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/ata.2014.v30.n1.3

Analysis in Theory and Applications, Vol. 30 (2014), Iss. 1 : pp. 34–50

Published online:    2014-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    17

Keywords:    Logarithmic differential forms de Rham complex regular meromorphic forms holonomic $\mathscr{D}$-modules Poincaré lemma mixed Hodge structure residual currents.