Vol. 318, No. 1, 2022

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Generalizations of degeneracy second main theorem and Schmidt's subspace theorem

Si Duc Quang

Vol. 318 (2022), No. 1, 153–188
DOI: 10.2140/pjm.2022.318.153
Abstract

By introducing the notion of distributive constant of a family of hypersurfaces with respect to a projective variety, we prove a second main theorem in Nevanlinna theory for meromorphic mappings with arbitrary families of hypersurfaces. Our second main theorem generalizes and improves previous results for meromorphic mappings with hypersurfaces, in particular for algebraically nondegenerate mappings with hypersurfaces in subgeneral position. The analogous results for the holomorphic curves with finite growth index from a complex disc into a projective variety, and for meromorphic mappings on a complete Kähler manifold are also given. For the last aim, we will prove a Schmidt’s subspace theorem for arbitrary families of homogeneous polynomials, which is the counterpart in the number theory of our second main theorem.

Keywords
Nevanlinna theory, diophantine approximation, second main theorem, subspace theorem, meromorphic mapping, hypersurface, homogeneous polynomial, subgeneral position
Mathematical Subject Classification
Primary: 11J68, 32H30
Secondary: 11J25, 30D35, 32A22
Milestones
Received: 23 January 2022
Revised: 13 March 2022
Accepted: 9 April 2022
Published: 1 August 2022
Authors
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 (TIMAS)
Thang Long University
Nghiem Xuan Yem, Hoang Mai
Hanoi
Vietnam