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Relative exactness modulo a polynomial map and algebraic $$({\mathbb C}^p,+)$$-actions. (English) Zbl 1067.14068
This article is a detailed version of the author’s paper [C. R. Acad. Sci., Paris, Sér. I, Math. 333, 557–560 (2001; Zbl 1037.14018)].
Let $$F=(f_1,\dots, f_q)$$ be a dominating polynomial map from $${\mathbb C}^n$$ to $${\mathbb C}^q$$ with $$n>q$$. Let $$\omega$$ be a polynomial differential 1-form on $${\mathbb C}^n$$. Then $$\omega$$ is called topologically relatively exact (abbreviated as TR-exact) if $$\omega$$ is exact along the generic fibers of $$F$$. If $$\omega$$ is a coboundary of the de Rham relative complex of $$F$$, namely, if $$\omega$$ is of type $$dR+a_1df_1+\dots +a_qdf_q$$ where $$R$$, $$a_i$$’s are polynomial functions on $${\mathbb C}^n$$, it is called algebraically relatively exact (abbreviated as AR-exact). The quotient module of TR-exact 1-forms by AR-exact 1-forms is denoted by $${\mathcal T}^1(F)$$. Note that $${\mathcal T}^1(F)$$ is a $${\mathbb C}[F]$$-module in a natural way, where $${\mathbb C}[F]$$ is the algebra generated by $$f_1,\dots, f_q$$. First, the author shows that $${\mathcal T}^1(F)$$ is a torsion $${\mathbb C}[F]$$-module. Next, TR-exact 1-forms are studied when $$F$$ is primitive, i.e., $$F$$ is a map such that $$\text{ codim}_{{\mathbb C}^q}B(F) \geq 1$$ and $$\text{ codim}_{{\mathbb C}^q}I(F)\geq 2$$, where $$B(F)=\{y \in {\mathbb C}^q \mid F^{-1}(y) \text{ is non-empty and not connected}\}$$ and $$I(F)=\{y \in {\mathbb C}^q \mid F^{-1}(y) \text{ is empty}\}$$.
Using the obtained results, it is shown that $${\mathcal T}^1(F)=0$$ if $$F$$ is quasi-fibered. The map $$F$$ is quasi-fibered iff (1) the set of singular points of $$F$$ is of codimension $$\geq 2$$, (2) $$\text{ codim}_{{\mathbb C}^q}B(F) \geq 2$$, and (3) $$\text{ codim}_{{\mathbb C}^q}I(F)\geq 3$$. The author states (without proofs) that a primitive mapping $$F$$ is quasi-fibered if and only if $${\mathcal T}^1(F)=0$$. Applying the techniques and tools used in the proofs of the above results to quotient morphisms by algebraic $$({\mathbb C}^p,+)$$-actions, some results are obtained:
Let $$\varphi : {\mathbb C}^p\times {\mathbb C}^n \to {\mathbb C}^n$$ be an algebraic $$({\mathbb C}^p,+)$$-action on $${\mathbb C}^n=\text{Spec\,} {\mathbb C}[x_1,\dots, x_n]$$ such that the invariant ring $${\mathbb C}[x_1,\dots, x_n]^{\varphi}$$ is a polynomial ring $${\mathbb C}[f_1,\dots, f_{n-p}]$$. Let $$F: {\mathbb C}^n \to {\mathbb C}^{n-p}:=\text{Spec\,} {\mathbb C}[x_1,\dots, x_n]^{\varphi}$$ be the quotient map. If $$F$$ is quasi-fibered and the set $${\mathcal NL}(\varphi)=\{x \in {\mathbb C}^n \mid$$ the orbit of $$x$$ is of dimension $$<p \}$$ has codimension $$\geq 2$$, then $$\varphi$$ is trivial, i.e., conjugate to a translation $$(x_1,\dots,x_n)\mapsto (x_1+t_1,\dots, x_p+t_p, x_{p+1},\dots, x_n)$$ for $$(t_1,\dots, t_p)\in {\mathbb C}^p$$. As a corollary, one obtains that every algebraic $$({\mathbb C}^{n-1},+)$$-action on $${\mathbb C}^n$$ such that $$\text{codim } {\mathcal NL}(\varphi)\geq 2$$ is trivial.
##### MSC:
 14R25 Affine fibrations 14R20 Group actions on affine varieties
##### Keywords:
relative cohomology; additive group action
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