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

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.