Abstract
Since the great work on holomorphic curves into algebraic varieties intersecting hypersurfaces in general position established by Ru in 2009, recently there has been some developments on the second main theorem into algebraic varieties intersecting moving hypersurfaces targets. The main purpose of this paper is to give some interesting improvements of Ru’s second main theorem for moving hypersurfaces targets located in subgeneral position with index.
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Corvaja, P. and Zannier, U., On a general Thue’s equation, Amer. J. Math., 126(5), 2004, 1033–1055.
Dethloff, G. and Tan, T. V., A second main theorem for moving hypersurface targets, Houston J. Math., 37, 2011, 79–111.
Dethloff, G. and Tan, T. V., Holomorphic curves into algebraic varieties intersecting moving hypersurfaces targets, Acta Math. Vietnamica, 45, 2020, 291–308.
Evertse, J. H. and Ferretti, R. G., A generalization of the subspace theorem with polynomials of higher degree, Diophantine Approximation, Festschrift for Wolfgang Schmidt. R. F. Tichy, H. P. Schlikewei, K. Schmidt (eds), Proceedings of a conference in honor of Prof. Wolfgang Schmidt’s 70th birthday, held in Vienna, October 6–10, 2003, Springer-Verlag, 2008, 175–198.
Fujimoto, H., Non-integrated defect relation for meromorphic maps of complete Kähler manifolds into \({\mathbb{P}^{{N_1}}}(\mathbb{C}) \times \cdots \times {\mathbb{P}^{{N_k}}}(\mathbb{C})\), Japanese J. Math. New series, 11(2), 1985, 233–264.
Ji, Q. C., Yan, Q. M. and Yu, G. S., Holomorphic curves into algebraic varieties intersecting divisors in subgeneral position, Math. Ann., 373, 2019, 1457–1483.
Noguchi, J. and Winkelmann, J., Nevanlinna Theory in Several Complex Variables and Diophantine Approximation, Springer-Verlag, Tokyo, 2014.
Quang, S. D., Second main theorem for meromorphic mappings with moving hypersurfaces in subgeneral position, J. Math. Anal. Appl., 465(1), 2018, 604–623.
Quang, S. D., Degeneracy second main theorems for meromorphic mappings into projective varieties with hypersurfaces, Trans. Amer. Math. Soc., 371(4), 2019, 2431–2453.
Ru, M., On the general form of the second main theorem, Trans. Amer. Math. Soc., 349, 1997, 5093–5105.
Ru, M., Nevanlinna Theory and its Relation to Diophantine Approximation, World Scientific Publishing Co., Singapore, 2001.
Ru, M., A defect relation for holomorphic curves intersecting hypersurfaces, Amer. J. Math., 126(1), 2004, 215–226.
Ru, M., Holomorphic curves into algebraic varieties, Ann. Math., 169, 2009, 255–267.
Shiffman, B., On holomorphic curves and meromorphic maps in projective space, Indiana Univ. Math. J., 28, 1979, 627–641.
Sombra, M., Bounds for the Hilbert function of polynomial ideals and for the degrees in the Nullstellensatz, Algorithms for algebra (Eindhoven, 1996). J. Pure Appl. Algebra, 117/118, 1997, 565–599.
Vojta, P., Diophantine Approximation and Nevanlinna Theory, Arithmetic Geometry, Lecture Notesin Math. Vol. 2009, 111–224. Springer-Verlag, Berlin, 2011.
Yan, Q. M. and Yu, G. S., Cartan’s conjecture for moving hypersurfaces, Math. Z., 292(3–4), 2019, 1051–1067.
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The authors are very grateful to the anonymous referee for some valuable suggestions and comments.
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This work was supported by the National Natural Science Foundation of China (Nos. 11871260, 11461042).
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Xie, L., Cao, T. Second Main Theorem for Meromorphic Maps into Algebraic Varieties Intersecting Moving Hypersurfaces Targets. Chin. Ann. Math. Ser. B 42, 753–776 (2021). https://doi.org/10.1007/s11401-021-0289-y
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DOI: https://doi.org/10.1007/s11401-021-0289-y