Second main theorems with weighted counting functions and its applications (Q2304135)

\textit{M. Ru} established a second main theorem for nondegenerate meromorphic mappings from \(\mathbb C\) into \(\mathbb P^n(\mathbb C)\) intersecting moving hyperplanes with truncated (to level \(n\)) counting functions [Proc. Am. Math. Soc. 129, No. 9, 2701--2707 (2001; Zbl 0977.32013)]. In [\textit{Do Duc Thai} and the reviewer, Int. J. Math. 16, No. 8, 903--939 (2005; Zbl 1086.32017)] this result was proved for the general case of meromorphic mappings from \(\mathbb C^m\). For the case of degenerate meromorphic mappings, \textit{M. Ru} and \textit{J. T. Y. Wang} [Trans. Am. Math. Soc. 356, No. 2, 557--571 (2004; Zbl 1042.32007)] gave a second main theorem for moving hyperplanes with a counting function truncated to level \(n\). Then this result of Ru and Wang was improved by \textit{D. D. Thai} and the reviewer in [Forum Math. 20, No. 1, 163--179 (2008; Zbl 1153.32019)] and the reviewer in [Abh. Math. Semin. Univ. Hamb. 86, No. 1, 1--18 (2016; Zbl 1360.32012)]. In [Proc. Am. Math. Soc. 144, No. 10, 4329--4340 (2016; Zbl 1345.32014)], the reviewer gave a new kind of second main theorem, called second main theorems with weighted counting functions, where each counting function of the divisor has a different coefficient, i.e, the role of the moving hyperplanes in the second main theorem may be different. In [Proc. Am. Math. Soc. 147, No. 4, 1657--1669 (2019; Zbl 1495.32045)], the reviewer gave some new second main theorem for meromorphic mappings and moving hyperplanes, which is stronger than all previous results in this area. In the paper under review, the authors use the idea of weighted counting functions from [Zbl 1345.32014)] to extend the result of [Zbl 1495.32045)]. The main result is the following. Theorem 1.1. Let \(f: \mathbb C^m\to\mathbb P^n(\mathbb C)\) be a meromorphic mapping. Let \(\{a_j\}_{j=1}^q\)\ \((q\geq 2n-k+2)\) be meromorphic mappings of \(\mathbb C^m\) into \(\mathbb P^n(\mathbb C)^*\) in general position such that \((f,a_j)\not\equiv0\)\ \((1\leq j\leq q)\). Let \(\lambda_1,...,\lambda_q\) be \(q\) positive numbers with \((2n-k+2)\max_{1\le i\le q}\lambda_i\le \sum_{i=1}^q\lambda_i\). Then for every positive number \(\eta\in [\max_{1\le i\le q}\lambda_i,\frac{\sum_{i=1}^q\lambda_i}{2n-k+2}]\), we have \[\big|\big|\ \frac{\sum_{j=1}^q\lambda_j-(n-k)\eta}{n+2}T_f(r)\leq\sum_{j=1}^q\lambda_{j}N^{[k]}_{(f,a_{j})}(r)+o(T_f(r))+O\left(\max\limits_{1\leq i\leq q}T_{a_i}(r)\right).\] As an application, an algebraic dependence theorem for meromorphic mappings into \(\mathbb P^n(\mathbb C)\) sharing moving hyperplanes is given.