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


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)


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 $\mathbb{R}^4$.


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

2010 Mathematics Subject Classification

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

G. Ishikawa was supported by JSPS KAKENHI no. 19K03458.

Received 9 December 2019

Accepted 26 February 2020

Published 17 February 2021