# Second Main Theorem and Unicity of Meromorphic Mappings for Hypersurfaces in Projective Varieties

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## 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 2*n*+3 hyperplanes in general position to the case where the mappings may be linearly degenerated.

### Keywords

Holomorphic curves Algebraic degeneracy Defect relation Nochka weight### Mathematics Subject Classification (2010)

Primary 32H30 Secondary 32H04 32H25 14J70## Notes

### Acknowledgments

This work was completed while the first author was staying at the Vietnam Institute for Advanced Study in Mathematics (VIASM). He would like to thank the Institute for the support.

This research is funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.04-2015.03.

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