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Relative cohomology of polynomial mappings

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Published in Manuscripta Mathematica. 2003, vol. 110, no. 4, p. 413-432
Abstract Let $F$ be a polynomial mappping from $mathbb{C}^n$ to $mathbb{C}^q$ with $n>q$. We study the De Rham cohomology of its fibres and its relative cohomology groups, by introducing a special fibre $F^{-1}(infty)$ "at infinity" and its cohomology. Let us fix a weighted homogeneous degree on $mathbb{C}[x_1,...,x_n]$ with strictly positive weights. The fibre at infinity is the zero set of the leading terms of the coordinate functions of $F$. We introduce the cohomology groups $H^k(F^{-1}(infty))$ of $F$ at infinity. These groups enable us to compute all the other cohomology groups of $F$. For instance, if the fibre at infinity has an isolated singularity at the origin, we prove that every weighted homogeneous basis of $H^{n-q}(F^{-1}(infty))$ is a basis of all the groups $H^{n-q}(F^{-1}(y))$ and also a basis a the $(n-q)^{th}$ relative cohomology group of $F$. Moreover the dimension of $H^{n-q}(F^{-1}(infty))$ is given by a global Milnor number of $F$, which only depends on the leading terms of the coordinate functions of $F$.
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BONNET, Philippe. Relative cohomology of polynomial mappings. In: Manuscripta Mathematica, 2003, vol. 110, n° 4, p. 413-432. https://archive-ouverte.unige.ch/unige:12295

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Deposited on : 2010-11-02

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