Partial properness and the Jacobian conjecture
The author, using his own results [ibid. 9, No. 1, 27-32 (1996; Zbl 0857.57022)], proves the following theorem on injectivity of regular maps: Let \(L,M\) and \(N\) be connected complex algebraic manifolds, \(M,N\) simply connected, and \(h=(f,g)\): \(N \mapsto L \times M\) be a regular local homeomorphism. If \(\lambda\) is a connected component of a fiber \(f^{-1}(u)\), \(u \in L\), then \(h\) is a homeomorphism if and only if \(g|\lambda\) is proper and \(f\) is proper on all the connected components of fibers of \(g\) that intersect \(\lambda\). The author also gives applications to the two-dimensional Jacobian conjecture \((L=M=N=\mathbb{C})\) for \(F =(P,Q)\): \(\mathbb{C}^2 \mapsto \mathbb{C}^2\), \(P,Q\) polynomials, \(\text{Jac} (F) \equiv 1\). One of them is a generalization of the result by \textit{J. Chadzyński} and the reviewer [Bull. Soc. Sci. Lett. Łódź, Sér. Rech. Déform. 14(1992/1993), No. 131-140, 13-19 (1993)] that if \(Q\) is proper on a fiber \(P^{-1} (u)\), \(u\in \mathbb{C}\), then \(F\) is a polynomial automorphism. The author shows that it suffices to assume that \(Q\) is proper on only one connected component of \(P^{-1} (u)\) (the last statement can be reduced to the above mentioned result because one can easily show that \(Q\) must be proper on the remaining connected components).
scientific article; zbMATH DE number 915066
scientific article