On holomorphic curves extremal for the truncated defect relation and some applications (Q815145)

The paper under review deals with extremal holomorphic curves for the truncated defect relation. Let \(f:\mathbb C\to \mathbb P_n(\mathbb C)\) be a transcendental linearly nondegenerate holomorphic curve and \(X\) a subset of \(\mathbb C^{n+1}\setminus \{0\}\) in \(N\)-subgeneral position. Denote by \(\delta_n({\mathbf a},f)\) the \(n\)-truncated defect of \(f\) at \(\mathbf a\). We define \(D^+_n(X,f) = \{\mathbf a\in X|\delta_n(\mathbf a,f) > 0\}\) and \(D^1_n(X, f) = \{\mathbf a\in X|\delta_n(\mathbf a,f) =1\}\). If there are \(n+1\) linearly independent vectors in \(D^1_n(X,f)\), then \(\#D^+_n(X,f)\leq (n + 1)(N + 1-n)\) (Theorem 3.1). The main purpose in this paper is to give estimate for the cardinality of \(D^1_n(X,f)\) when \(f\) is extremal for the truncated defect relation. Suppose that \(\sum_{\mathbf a_j\in D^1_n(X,f)}\delta(\mathbf a_j,f)= 2N-n+1\). If there are \(n + 1\) linearly independent vectors in \(D^1_n(X,f)\), then \(D^+_n(X,f)=D^1_n(X,f)\) and \(\#D^1_n(X,f)=2N-n + 1\) (Theorem 3.2). When there are \(n\) linearly independent vectors in \(D^1_n(X,f)\), if \(\# D^1_n(X,f) < 2N-n+1\), then \(\#D^1_n(X,f) = N\) (Theorem 3.3). The author also considers another defect and obtains the results similar to the above.