A defect relation for holomorphic curves intersecting general divisors in projective varieties
Let \(D\) be an effective Cartier divisor on a normal projective variety \(X\). In this paper, the notion of its Nevanlinna constant, denoted by Nev\,\((D)\), is introduced. Nev\,\((D)\) is given by \(\text{Nev}\,(D):=\inf_N\left(\inf_{\{\mu_N,V_N\}}\frac{\dim\,V_N}{\mu_N}\right)\), where the infimum ``\(\inf_N\)'' is taken over all positive integers \(N\) and the infimum ``\(\inf_{\{\mu_N,V_N\}}\)'' is taken over all pairs \({\{\mu_N,V_N\}}\) where \(\mu_N\) is a positive real number and \(V_N\subset H^0(X,\mathcal{O}(ND))\) is a linear subspace with \(\dim V_N\geq2\) such that, for all \(P\in\text{supp}\,D\), there exists a basis \(B\) of \(V_N\) with \[ \sum_{s\in B}\,\text{ord}_E(s)\geq\mu_N\,\text{ord}_E(ND) \] for all irreducible components \(E\) of \(D\) passing through \(P\). If \(\dim H^0(X,\mathcal{O}(ND))\leq1\) for all positive integers \(N\), then \(\text{Nev\,}(D):=+\infty\). Let \[ \delta_f(D):=\liminf_{r\to\infty}\frac{m_f(r,D)}{T_{f,D}(r)} \] be the Nevanlinna defect of \(f\) with respect to \(D\). A defect relation \(\delta_f(D)\leq\,\text{Nev}\, (D)\) for any Zariski-dense holomorphic mapping \(f:\mathbb{C}\rightarrow X\) is proved. It gives a unified proof (by simply computing \(\text{Nev\,}(D)\)) of previous results of the author (2004, 2009). More importantly, it also gives a new defect relation for holomorphic mappings \(f:\mathbb{C}\rightarrow X\) intersecting divisors \(D_1,\dots, D_q\), where \(D_1,\dots, D_q\) are not necessarily linearly equivalent. One has \(\delta_f(D')<\frac{2l\,\dim\,X}{q}\) with \(D':=\sum_{i=1}^qr_iD_i\), where \(D_1+\cdots+D_q\) is a sum of big and nef Cartier divisors, in \(l\)-subgeneral position on \(X\), and \(r_i>0\) are real numbers such that \(D'\) is equidegree.
scientific article; zbMATH DE number 6658543
scientific article