No Access

Degeneracy second main theorem for meromorphic mappings and moving hypersurfaces with truncated counting functions and applications

Si Duc Quang

Si Duc Quang

Department of Mathematics, Hanoi National University of Education, 136-Xuan Thuy, Cau Giay, Hanoi, Vietnam

Thang Long Institute of Mathematics and Applied Sciences, Nghiem Xuan Yem, Hoang Mai, Hanoi, Vietnam

E-mail Address: [email protected]

Search for more papers by this author

https://doi.org/10.1142/S0129167X20500457Cited by:2 (Source: Crossref)

Abstract

In this paper, we establish a new second main theorem for meromorphic mappings of $ℂ^{m}$ into $ℙ^{n} (ℂ)$ and moving hypersurfaces with truncated counting functions in the case, where the meromorphic mappings may be algebraically degenerate. A version of the second main theorem with weighted counting functions is also given. Our results improve the recent results on this topic. As an application, an algebraic dependence theorem for meromorphic mappings sharing moving hypersurfaces is given.

Keywords:

AMSC: Primary: 32H30, Secondary: 30D35, Secondary: 32A22

References

1. T. T. H. An and H. T. Phuong, An explicit estimate on multiplicity truncation in the second main theorem for holomorphic curves encountering hypersurfaces in general position in projective space, Houston J. Math. 35 (2009) 775–786. Web of Science, Google Scholar
2. T. B. Cao and H. Z. Cao, Multiple values and uniqueness problem of meromorphic mappings sharing hypersurfaces, Complex Var. Elliptic Equ. 62 (2017) 520–535. Crossref, Web of Science, Google Scholar
3. G. Dethloff and T. V. Tan, A second main theorem for moving hypersurface targets, Houston J. Math. 37 (2011) 79–111. Web of Science, Google Scholar
4. G. Dethloff and T. V. Tan, A uniqueness theorem for meromorphic maps with moving hypersurfaces, Publ. Math. Debrecen 78 (2011) 347–357. Crossref, Web of Science, Google Scholar
5. H. Fujimoto, Non-integrated defect relation for meromorphic maps of complete Kähler manifolds into $ℙ^{N_{1}} (ℂ) \times \dots \times ℙ^{N_{k}} (ℂ)$ , Japanese J. Math. 11 (1985) 233–264. Crossref, Google Scholar
6. J. Noguchi and T. Ochiai, Introduction to Geometric Function Theory in Several Complex Variables, Translated Mathematical Monography, Vol. 80 (American Mathematical Society, Providence, Rhode Island, 1990). Crossref, Google Scholar
7. S. D. Quang, Second main theorem and unicity theorem for meromorphic mappings sharing moving hypersurfaces regardless of multiplicity, Bull. Sci. Math. 136 (2012) 399–412. Crossref, Web of Science, Google Scholar
8. S. D. Quang, Second main theorems and unicity of meromorphic mappings with moving hypersurfaces, Bull. Math. Soc. Sci. Math. Roumanie 57(105(3)) (2014) 279–300. Google Scholar
9. S. D. Quang and D. P. An, Second main theorems for meromorphic mappings with moving hypersurfaces and uniqueness problem, Comput. Methods Funct. Theory 17 (2017) 445–461. Crossref, Web of Science, Google Scholar
10. S. D. Quang, Second main theorems for meromorphic mappings and moving hyperplanes with truncated counting functions, Proc. Amer. Math. Soc. 147 (2019) 1657–1669. Crossref, Web of Science, Google Scholar
11. S. D. Quang, Degeneracy second main theorems for meromorphic mappings into projective varieties with hypersurfaces, Trans. Amer. Math. Soc. 371(4) (2019) 2431–2453. Crossref, Web of Science, Google Scholar
12. S. D. Quang and D. P. An, Second main theorem and unicity of meromorphic mappings for hypersurfaces of projective varieties, Acta Math. Vietn. 42 (2017) 455–470. Crossref, Web of Science, Google Scholar
13. M. Ru, Geometric and arithmetic aspects of $ℙ^{n}$ minus hyperplanes, Amer. J. Math. 117 (1995) 307–321. Crossref, Web of Science, Google Scholar
14. M. Ru, A uniqueness theorem with moving targets without counting multiplicity, Proc. Amer. Math. Soc. 129 (2001) 2701–2707. Crossref, Web of Science, Google Scholar
15. M. Ru and J. T-Y. Wang, Truncated second main theorem with moving targets, Trans. Amer. Math. Soc. 356 (2004) 557–571. Crossref, Web of Science, Google Scholar
16. M. Ru, Holomorphic curves into intersecting hypersurfaces, Ann. Math. 169 (2009) 225–267. Crossref, Web of Science, Google Scholar
17. B. Shiffman, Introduction to the Carlson-Griffiths equidistribution theory, Lecture Notes in Math. 981 (1983) 44–89. Crossref, Web of Science, Google Scholar
18. W. Stoll, On the propagation of dependences, Pacific J. Math. 139 (1989) 311–337. Crossref, Web of Science, Google Scholar
19. D. D. Thai and S. D. Quang, Second main theorem with truncated counting function in several complex variables for moving targets, Forum Math. 20 (2008) 163–179. Crossref, Web of Science, Google Scholar
20. D. D. Thai, S. D. Quang and D. P. An, The second main theorem for meromorphic mappings into a complex projective space, Acta Math. Vietn. 38 (2013) 187–205. Crossref, Google Scholar