General defect relations of holomorphic curves

Abstract

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