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# 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

#### 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

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