Abstract:

Based on the derived holomorphic curves, the Milloux type inequality of holomorphic curves is established, and the Picard type theorem is proved: let $f$ be a holomorphic curve from $\mathbf{C}$ to $P^{n}(\mathbf{C}), \nabla f$ is a derived holomorphic curve of $f$ and $\left\{H_{j}\right\}_{j=1}^{2 n+1}$ be hyperplanes in $P^{n}(\mathbf{C})$ located in general position, if $f$ omits the hyperplane family $\left\{H_{j}\right\}_{j=1}^{n+1}, \nabla f$ omits the hyperplane family $\left\{H_{j}\right\}_{j=n+2}^{2 n+1}$, then $f$ is constant, where $H_{1}, H_{2}, \cdots, H_{n+1}$ are $n+1$ coordinate hyperplanes. And some examples are given to show that the number of hyperplanes cannot be reduced and the coordinate hyperplanes cannot be generalized to arbitrary hyperplanes when $n \geqslant 2$.

Key words: holomorphic curve, derived holomorphic curve, Picard type theorem