On a certain holomorphic curve extremal for the defect relation (Q2386857)

The paper under review concerns holomorphic curves in complex projective spaces with maximal sum of deficiencies. Let \(f: \mathbb{C}\to\mathbb{P}^n(\mathbb{C})\) be a transcendental linearly nondegenerate holomorphic curve with a reduced representation \((f_1,\dots, f_{n+1})\). Let \(X\) be a subset of \(\mathbb{C}^{n+1}\setminus\{0\}\) in \(N\)-subgeneral position \((N\geq n\geq 2)\). The defect relation due to Cartan-Nochka implies \(\sum^q_{j=1} \delta({\mathbf a}_j,f)\leq 2N- n+1\) for \({\mathbf a}_j\in X\) \((q> 2N- n+ 1)\). The present paper deals with the case when the defect sum is maximal, that is, \[ \sum^q_{j=1} \delta({\mathbf a}_j, f)= 2N- n+ 1,\tag{1} \] and the existence of \({\mathbf a}_j\in X\) with \(\delta({\mathbf a}_j,f)= 1\) is proved under some conditions. Let \(u_k(z)= \max\{|f_j(z)|: 1\leq j\leq n+1\), \(j\neq k\}\) and \[ t_k(r,f)= (1/2\pi) \int^{2\pi}_0 u_k(re^{i\theta})\,d\theta+ O(1). \] Define \(\Omega_k= \limsup_{r\to+\infty} t_k(r,f)/T(r,f)\). Then the author proves that if (1) holds and \(\Omega_k< 1\) for some \(k\), then there is a subset \(P\subset\{1,\dots, q\}\) with \(\sharp P= N- n+1\) such that \(\delta({\mathbf a}_j, f)= 1\) for \(j\in P\) and \(d(P)= 1\), where \(\sharp P\) denotes the cardinality and \(d(P)\) is the dimension of the linear span of \(\{{\mathbf a}_j: j\in P\}\) (Theorems 1 and 2). It is also proved that if \(f\) is an exponential curve and \(N> n\), then (1) does not hold (Theorem 3). For related results, see \textit{N. Toda} [Kodai Math. J. 24, No. 1, 134--146 (2001; Zbl 0982.32018)].