Second main theorem and unicity of meromorphic mappings for hypersurfaces in projective varieties

Abstract

Let $V$ be a projective subvariety of $\mathbb P^n(\mathbb C)$. A family of hypersurfaces $\{Q_i\}_{i=1}^q$ in $\mathbb P^n(\mathbb C)$ is said to be in $N$-subgeneral position with respect to $V$ if for any $1\le i_1<\cdots <i_{N+1}$, $ V\cap (\bigcap_{j=1}^{N+1}Q_{i_j})=\emptyset$. In this paper, we will prove a second main theorem for meromorphic mappings of $\mathbb C^m$ into $V$ intersecting hypersurfaces in subgeneral position with truncated counting functions. As an application of the above theorem, we give a uniqueness theorem for meromorphic mappings of $\mathbb C^m$ into $V$ sharing a few hypersurfaces without counting multiplicity. In particular, we extend the uniqueness theorem for linear nondegenerate meromorphic mappings of $\mathbb C^m$ into $\mathbb P^n(\mathbb C)$ sharing $2n+3$ hyperplanes in general position to the case where the mappings may be linear degenerate.