Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 019, 21 pages      arXiv:2007.09950
Contribution to the Special Issue on Primitive Forms and Related Topics in honor of Kyoji Saito for his 77th birthday

Computing Regular Meromorphic Differential Forms via Saito's Logarithmic Residues

Shinichi Tajima a and Katsusuke Nabeshima b
a) Graduate School of Science and Technology, Niigata University, 8050, Ikarashi 2-no-cho, Nishi-ku Niigata, Japan
b) Graduate School of Technology, Industrial and Social Sciences, Tokushima University, 2-1, Minamijosanjima-cho, Tokushima, Japan

Received July 24, 2020, in final form February 05, 2021; Published online February 27, 2021

Logarithmic differential forms and logarithmic vector fields associated to a hypersurface with an isolated singularity are considered in the context of computational complex analysis. As applications, based on the concept of torsion differential forms due to A.G. Aleksandrov, regular meromorphic differential forms introduced by D. Barlet and M. Kersken, and Brieskorn formulae on Gauss-Manin connections are investigated. $(i)$ A method is given to describe singular parts of regular meromorphic differential forms in terms of non-trivial logarithmic vector fields via Saito's logarithmic residues. The resulting algorithm is illustrated by using examples. $(ii)$ A new link between Brieskorn formulae and logarithmic vector fields is discovered and an expression that rewrites Brieskorn formulae in terms of non-trivial logarithmic vector fields is presented. A new effective method is described to compute non trivial logarithmic vector fields which are suitable for the computation of Gauss-Manin connections. Some examples are given for illustration.

Key words: logarithmic vector field; logarithmic residue; torsion module; local cohomology.

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