On the defect relation of holomorphic curves for moving targets (Q2712717)

Let \(f=[f_1, \dots,f_{n+1}]\) be a transcendental holomorphic curve from \(\mathbb C\) into the \(n\)-dimensional complex projective space \(\mathbb P^n(\mathbb C)\), \(T(r, f)\) the characteristic function of \(f\), \(X\) a subset of \(\mathbb C^{n+1}-\{0\}\) in \(N\)-subgeneral position, where \(N \geq n\) are positive integers, and \(X(0)=\{a = (a_1, \dots, a_{n+1})\in X| a_{n+1}=0\}\). Put NEWLINE\[NEWLINEt(r, f) = {1\over 2\pi} \int_0^{2\pi} \{\log \max_{1 \leq j \leq n}|f_j(re^{i\theta})| - \log\max_{1 \leq j \leq n}|f_j(e^{i\theta})|\}\,d\theta.NEWLINE\]NEWLINENEWLINENEWLINEIn an earlier paper [Proceedings of the Second ISAAC Congress, Kluwer, Dordrecht, Int. Soc. Anal. Appl. Comput. 7, 501--510 (2000; Zbl 1037.32019)] the author proved the following result: For any \(a_1, \dots, a_q \in X (2N-n+1 < q < \infty),\) NEWLINE\[NEWLINE\sum_{j=1}^q \omega(j) \delta(a_j, f)\leq d+1+(n-d)\Omega,NEWLINE\]NEWLINE where \(\omega\) is a Nochka weight function for \(a_1, \dots, a_q\), \(d = \sum_{a_j \in X(0)} \omega(j)\) and \newline NEWLINE\(\Omega = \limsup_{r \rightarrow \infty} t(r, f)/T(r, f).\)NEWLINENEWLINEThe paper under review extends the above result to moving targets, using the technique of Ru-Stoll. This also gives an improvement of Ru-Stoll's result regarding the second main theorem with moving targets.