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