General defect relations of holomorphic curves

Abstract:

Let $x:{\mathbf {C}} \to {P_n}{\mathbf {C}}$ be a holomorphic curve of finite lower order $\mu$, and let $A = \{ \alpha \}$ be an arbitrary finite family of holomorphic curves $\alpha :{\mathbf {C}} \to {({P_n}{\mathbf {C}})^\ast }$ satisfying $T(r,\alpha ) = o(T(r,x))\;(r \to \infty )$. Suppose $x$ is nondegenerate with respect to $A$, and $A$ is in general position. We show the following general defect relations: (1) $x$ has at most $n$ deficient curves in $A$ if $\mu = 0$. (2) $\sum \nolimits _{\alpha \in A} {\delta (\alpha ) \leq n\;{\text {if}}\;0 < \mu \leq 1/2}$. (3) $\sum \nolimits _{\alpha \in A} {\delta (\alpha ) \leq [2n\mu ] + 1\;{\text {if}}\;1/2 < \mu < + \infty }$.