Pure and Applied Mathematics Quarterly

Volume 16 (2020)

Number 5

Singular mappings and their zero-forms

Pages: 1619 – 1633

DOI: https://dx.doi.org/10.4310/PAMQ.2020.v16.n5.a9

Authors

Goo Ishikawa (Faculty of Sciences, Department of Mathematics, Hokkaido University, Sapporo, Japan)

Stanisław Janeczko (Institute of Mathematics, Polish Academy of Sciences, Warszawa, Poland; and Warsaw University of Technology, Faculty of Mathematics and Information Science, Warszawa, Poland)

Abstract

We study the quotient complexes of the de Rham complex on singular mappings; the complex of algebraic restrictions, the complex of geometric restrictions and the residual complex. Vanishing theorem for algebraic, geometric and residual cohomologies on quasi-homogeneous map-germs was proved. The finite order and symplectic zero-forms were characterized on parametric singularities. In this context the singular parametric Lagrangian surfaces were investigated, with the classification list of $\mathcal{A}$-simple Lagrangian singularities of $\mathbb{R}^2$ into $\mathbb{R}^4$.

Keywords

differential forms, singularities, geometric restriction, algebraic restriction, residual cohomology, parametric curves and surfaces

2010 Mathematics Subject Classification

Primary 53D05. Secondary 57R42, 58A10, 58K05.

Received 9 December 2019

Accepted 26 February 2020

Published 17 February 2021