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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.
14R25 Affine fibrations
14R20 Group actions on affine varieties
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