On families of meromorphic maps into the complex projective space (Q2834124)

The authors prove some normality criteria for families of holomorphic mappings of several complex variables into an arbitrary closed set in \(\mathbb P^N(\mathbb C)\) (with respect to the usual topology of a real manifold of dimension \(2n\)). As applications, some meromorphic normality criteria for families of meromorphic mappings are given.NEWLINENEWLINETheorem 1. Let \(X\) be a closed subset of \(\mathbb P^N(\mathbb C).\) Let \(\mathscr F\) and \(\mathscr G\) be two families of holomorphic mappings of a domain \( D \subset\mathbb C^m\) into \(X\). Let \(Q_1, \ldots , Q_{3t+1}\) be hypersurfaces in \(\mathbb P^N(\mathbb C)\) located in \(t\)-subgeneral position with respect to \(X\). If \(\mathscr G\) is normal, and for every \(f \in \mathscr F,\) there exists \(g \in {\mathscr G}\) such that \(f^{-1}(Q_j)=g^{-1}(Q_j)\) (\(j=1,\ldots, 3t + 1\)), then \(\mathscr F\) is normal. NEWLINENEWLINENEWLINENEWLINE Theorem 2. Let \(X\) be a closed subset of \(\mathbb P^N(\mathbb C)\). Let \(\mathscr F\) and \(\mathscr G\) be two families of holomorphic mappings of a domain \( D \subset \mathbb C^m\) into \(X\). Let \(Q_1, \ldots , Q_{3t+1}\) be hypersurfaces in \(\mathbb P^N(\mathbb C)\) located in \(t\)-subgeneral position with respect to \(X\). Assume that NEWLINENEWLINENEWLINENEWLINE \(i)\) for any fixed compact subset \(K \subset D\), all \(f \in \mathscr F\) and \(i = 1,..., t + 1\), the Lebesgue area of \(f^{-1}(Q_i)\cap K\) is bounded above and \(f(D) \not \subseteq Q_i\), NEWLINENEWLINENEWLINENEWLINE \(ii)\) for every \(f \in \mathscr F\), there exists \(g \in \mathscr G\) such that \(f^{-1}(Q_j)=g^{-1}(Q_j)\) (\(j=1,\ldots, 3t + 1\)), NEWLINENEWLINENEWLINENEWLINE \(iii)\) \(f\) is meromorphically normal. NEWLINENEWLINENEWLINENEWLINE Then \(\mathscr F\) is also meromorphically normal. NEWLINENEWLINENEWLINENEWLINE The authors also improve Fujimoto's theorem as follows.NEWLINENEWLINENEWLINETheorem 3. Let \(\mathscr F\) be a family of meromorphic mappings of a domain \(D \subset\mathbb C^m\) into \(\mathbb P^N(\mathbb C)\), and let \(H_1,\ldots, H_{2t+1}\) be hyperplanes in \(\mathbb P^N(\mathbb C)\) located in \(t\)-subgeneral position such that for each \(f \in \mathscr F\), \(f(D) \not \subseteq H_j\), (\( j = 1,..., 2t+1)\), and for any fixed compact subset \(K \subset D\), the Lebesgue areas of \(f^{-1}(H_j)\cap K\) (\(j = 1,\dots, 2t + 1\)) are bounded above, for all \(f \in \mathscr F\). Then \( \mathscr F\) is meromorphically normal.