Asian Journal of Mathematics

Volume 22 (2018)

Number 6

On singular varieties associated to a polynomial mapping from $\mathbb{C}^n$ to $\mathbb{C}^{n-1}$

Pages: 1157 – 1172

DOI: https://dx.doi.org/10.4310/AJM.2018.v22.n6.a9

Authors

Nguyen Thi Bich Thuy (Ibilce-Unesp, Universidade Estadual Paulista, Instituto de Biociências, Letras e Ciências Exatas, São José do Rio Preto, Brazil)

Maria Aparecida Soares Ruas (Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Brazil)

Abstract

We construct singular varieties $\mathcal{V}_G$ associated to a polynomial mapping $G : \mathbb{C}^n \to \mathbb{C}^{n-1}$ where $n \geqslant 2$. Let $G : \mathbb{C}^3 \to \mathbb{C}^2$ be a local submersion, we prove that if the homology or the intersection homology with total perversity (with compact supports or closed supports) in dimension two of any variety $\mathcal{V}_G$ is trivial then $G$ is a fibration. In the case of a local submersion $G : \mathbb{C}^n \to \mathbb{C}^{n-1}$ where $n \geqslant 4$, the result is still true with an additional condition.

Keywords

complex polynomial mappings, intersection homology, singularities at infinity

2010 Mathematics Subject Classification

14P10, 14R15, 32S20, 55N33

Full Text (PDF format)

The research was partially supported by the post-doctoral FAPESP 2013/18706-7 (for the first author) and FAPESP Proc. 2014/00304-2 and CNPq Proc. 305651/2011-0 (for the second author).

Received 16 March 2016

Accepted 5 April 2017

Published 6 February 2019