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≤i ₁<⋯<i _N+1≤q, $V\cap (\bigcap _{j=1}^{N+1}Q_{i_{j}})=\varnothing$. 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 linearly 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 linearly degenerated.