Relative cohomology of polynomial mappings
The author studies polynomial maps \(F:\mathbb{C}^n \to\mathbb{C}^q\), \(n>q\), that are dominating, i.e. the image of \(F\) is dense in \(\mathbb{C}^q\) in the Zariski topology. This means, for \(F=(f_1,f_2, \dots,f_q)\), \(\mathbb{C}[F]= \mathbb{C} [f_1,f_2, \dots, f_q]=F^* (\mathbb{C}[t_1, t_2,\dots, t_q])\). In \(\mathbb{C} [x_1,x_2, \dots, x_n]\), a positive weighted degree is introduced by the weights \(p_1>0\), \(p_2>0, \dots, p_n>0\) assigned to \(x_1,x_2, \dots,x_n\). The leading terms \(\overline f_1,\overline f_2,\dots, \overline f_q\) are used to construct a fiber ``at infinity'', \(F^{-1} (\infty)\), and to introduce (de Rham) cohomology groups \(H^k(F^{-1} (\infty))\). Then the cohomology groups of the fiber \(F^{-1}(y) \), \(y\in \mathbb{C}^q\), are considered. In particular, \(H^k(F^{-1} (y))=0\), \(y\in \mathbb{C}^q\), for \(k\) small enough: \(0<k<n-q -\dim\text{Sing}(F^{-1} (\infty))\). Also, if \(F^{-1} (\infty)\) has an isolated singularity at the origin, then any quasi-homogeneous basis of \(H^{n-q}(F^{-1} (\infty))\) provides a basis for \(H^{n-q} (F^{-1} (y))\), \(y\in \mathbb{C}^q\), as well as a basis of the \((n-q)\)th relative cohomology group of \(F\). Moreover, the dimension of all these groups is equal to a global Milnor number of \(F\), which only depends on \(\overline f_1, \overline f_2, \dots,\overline f_q\).
scientific article; zbMATH DE number 1623443