Second main theorems for meromorphic mappings with moving hypersurfaces and a uniqueness problem
The paper deals with the second main theorem for meromorphic mappings and moving hypersurfaces in complex projective space. Let \(f:{\mathbb C}^m \to{\mathbb P}^n({\mathbb C})\) be a meromorphic mapping. Let \(\{Q_j\}_{j=1}^q\) be a family of slowly moving hypersurfaces in weak general position in \({\mathbb P}^n({\mathbb C})\) with \(\deg Q_j = d_j\). Assume that \(Q_j(f)\not\equiv 0\) for all \(j\). Set \(N ={ {n+d}\choose{n}}-1\), where \(d\) is the greatest common divisor of \(d_1,\ldots,d_q\). Then the authors give some inequalities of second main theorem-type for \(f\). For instance, they prove the following. If \(q \geq nN + n + 1\), then \[ \frac{q -(n-1)(N + 1)}{N + 2} T_f (r) \leq \sum_{j=1}^q \frac{1}{d_j} N^{[N]}_{Q_j (f)}(r) + o(T_f (r)) + O(\max_j T_{Q_j}(r)). \] They also give some uniqueness theorems for \(f\) and \(\{Q_j\}_{j=1}^q\).
scientific article; zbMATH DE number 6821106
scientific article