On holomorphic curves extremal for the \(\mu _{n}\)-defect relation (Q2372844)

Let \(f\) be a transcendental linearly nondegenerate holomorphic curve from \(\mathbb{C}\) to the \(n\)-dimensional complex projective space \(\mathbb C P^n\) with a reduced representation \((f_1, \dots, f_{n+1})\). A vector \(a=(a_1, \dots, a_{n+1})\in \mathbb{C}^{n+1}\setminus \{0\}\) has multiplicity \(m\) if the entire function \(\sum a_jf_j\) has at least one zero and all the zeros of \(\sum a_jf_j\) have multiplicity at least \(m\). When \(\sum a_jf_j\) has no zero, we set \(m=\infty\). Further, for a positive integer \(k\), the \(\mu_k\)-defect of \(a\) with respect to \(f\) is defined by \(\mu_k (a, f)=1-k/\max(m, k)\). Let \(X\) be any subset of \(\mathbb{C}^{n+1}\setminus \{0\}\) in \(N\)-subgeneral position satisfying \(2N-n+2\leq \#X \leq \infty\). The \(\mu_k\)-defect relation for \(f\) over \(X\) states that, for any \(a_1, \dots, a_q \in X\), \(\sum \mu_n(a_j, f)\leq 2N-n+1\). In this paper the author proves several results on the \(\mu_k\)-defect relation for \(f\) and on \(\mu_k (a, f)\) when the \(\mu_k\)-defect relation reaches extremality. For example, let \(M^1_k(X, f)=\{a\in X: \mu_n(a, f)=1\}\), the author proves that if there exists \(a\in X\setminus M^1_k(X, f)\) satisfying \(\delta(a, f)>0\), then the \(\mu_k\)-defect relation is not extremal. In sections four and five, the author also shows some results when the \(\mu_k\)-defect relation is extremal, \(n=2m\) or \(n=2m-1\). Several interesting examples are given in the end.