Logarithmic differential forms, torsion differentials and residue

Abstract

This article deals with the theory of logarithmic differential forms and with some of its less known applications, developed by the author in the past few years, in complex analysis, topology and geometry of singular varieties and in the theory of differential equations. At first, some relations between properties of logarithmic differential forms and torsion differentials on singular hypersurfaces are studied. Then a natural extension of the classical Poincaré residue in the case of singular hypersurfaces, and a variant of the Grothendieck local duality for non-isolated non-normal reduced hypersurface singularities are described in detail. In conclusion a new method for computing the index of vector fields on hypersurfaces with arbitrary singularities, and some applications to the theory of Pfaffian systems of differential equations of Fuchsian type are discussed.

Keywords:

• Logarithmic differential forms
• Torsion differentials
• Regular meromorphic differential forms
• Poincaré–Leray residue
• Logarithmic residue
• Duality in tangent cohomology
• Index of vector fields
• Fuchsian systems
• AMS 2000 Mathematics Subject Classifications: 32S25
• 14F10
• 14F40
• 35N05
• 58K65

Acknowledgement

The author is deeply indebted to Professor Heinrich Begehr for providing an opportunity to present and discuss part of this work during his seminars on complex, hypercomplex and Clifford analysis held at Freie Universität Berlin over many years.

Notes

Email: [email protected]

Dedicated to Professor Heinrich Begehr on the occasion of his 65th birthday.

Dedicated to Professor Heinrich Begehr on the occasion of his 65th birthday.

Additional information

Notes on contributors

Aleksandr G. Aleksandrov *

Email: [email protected] Dedicated to Professor Heinrich Begehr on the occasion of his 65th birthday.