Multidimensional residue theory and the logarithmic de Rham complex

A. G. Aleksandrov

Journal of Singularities
volume 5 (2012), 1-18
Proceedings of the International Conference on Singularity Theory and Applications, Hefei, China, July 25-31, 2011

Received 22 December 2011. Received in revised form 10 May 2012.

DOI: 10.5427/jsing.2012.5a

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We study logarithmic differential forms with poles along a reducible hypersurface and the multiple residue map with respect to the complete intersection given by its components. Some applications concerning computation of the kernel and image of the residue map and the description of the weight filtration on the logarithmic de Rham complex for hypersurfaces whose irreducible components are defined by a regular sequence of functions are considered. Among other things we give an easy proof of the de Rham theorem (1954) on residues of closed meromorphic differential forms whose polar divisor has rational quadratic singularities, and correct some inaccuracies in its original formulation and later citations.


logarithmic de Rham complex; regular meromorphic forms; multiple residues; complete intersections; weight filtration

Author(s) information:

A. G. Aleksandrov
Institute for Control Sciences
Russian Academy of Sciences
Profsojuznaja str. 65, B-334, Moscow
117997, Russian Federation