Geometrical and topological properties of real polynomial fibres (Q1827415)

Let \(f:\mathbb{R}^n\to\mathbb{R}\) be a polynomial having 0 as a regular value, and let \(k\) be the Gauss curvature of the \((n-1)\)-dimensional manifold \(V=f^{-1}(0)\) oriented by the gradient of \(f\). The author proves the existence of the integral \(\int_Vk\,dv\) as the limit \[ \int_Vk\,dv= \lim_{r\to\infty} \int_{K_r}k\,dv, \] where \(\{K_r\}_{r>0}\) is a sequence of compact sets such that \(V=\Cup K_r\) and \(K_r \subset K_{r'}\) if \(r< r'\), and shows the independence of this limit of the choice of the sequence \(\{K_r\}_{r>0}\). The main result of the paper under review is Theorem 4.5, which provides an explicit formula for this integral. Such a formula extends to the algebraic noncompact case the classical Gauss-Bonnet formula due to \textit{A. Haeflinger} [Ann. Inst. Fourier 10, 47--60 (1960; Zbl 0095.37704)] and \textit{H. Samelson} [Can. J. Math. 12, 529--534 (1960; Zbl 0099.39302)] if \(n\) is odd, and \textit{H. Hopf} [``Differential Geometry in the Large'', Seminar Lectures New York University 1946 and Stanford University 1956. Lect. Notes Math. 1000 (1983; Zbl 0526.53002)] if \(n\) is even. Explicitely, let us denote \(du\) the volume form of \(\mathbb{S}^{n-1}\) and for each \(u\in\mathbb{S}^{n-1}\), the projection \(P_u:\mathbb{R}^n\to\mathbb{R}:x\mapsto\langle u,x\rangle.\) Then, if \(n\) is odd, \[ \int_Vk\,dv=\frac 12\text{Vol}(\mathbb{S}^{n-1})\chi(V)-\frac 12 \int_{\mathbb{S}^{n-1}} \chi (V\cap \{P_u=0\})\,du \] (there is a sign mistake in line 10 of page 44 of the paper), while for even \(n\) it is proved that \[ \begin{multlined}\int_V k\,dv= \frac 12\text{Vol}(\mathbb{S}^{n-1}) [\chi (\{f\geq 0\})- \chi( \{f\leq 0\})]+\\ +\frac 12 \int_{ \mathbb{S}^{n-1}} [\chi(\{f\geq 0\}\cap\{P_u=0\})-\chi (\{f \leq 0\}\cap\{P_u=0\})] \,du.\end{multlined} \] In spite of the depth of the result, the proof is very clear and elegant. It combines techniques and results coming from several mathematical fields: differential geometry and topology, classical algebraic geometry and real algebraic geometry. The key point in the proof is a lemma which has its own interest: it says that for a generic \(u\in\mathbb{S}^{n-1}\) the restriction to \(V\) of the projection \(P_u\) does not have singularities at infinity, and that a certain linear combination of Euler characteristics of sets involving \(V\), \(\{f\geq 0\}\), \(\{f\leq 0\}\), \(P_u=c\), and a regular value \(c\) of \(P_u\) does not depend on it.