Some Researches On The Value Distribution Theory Of Holomorphic Curves

The theory on value distribution of holomorphic curves is a significant component of complex analysis theory.It exhibits strong connections with mathematical fields such as differential geometry,algebraic geometry,and number theory.As a research tool,its development has significantly advanced various fields of complex analysis,including complex differential geometry and complex dynamical systems.In recent decades,scholars from around the world,including domestic researchers,have made efforts to simplify the conditions of the second main theorem in the theory of value distribution of holomorphic curves.Consequently,the theory of value distribution of holomorphic curves has emerged as a new and popular research direction.This thesis focuses on the value distribution problem of high-dimensional holomorphic curves and meromorphic mappings.It presents several second main theorems concerning holomorphic curves and meromorphic mappings under different conditions,accompanied by relevant applications.The thesis is divided into seven chapters.Chapter 1 presents a comprehensive overview of the current research status of the theory of value distribution of holomorphic curves,covering both domestic and international perspectives.It examines the research objectives,significance,and fundamental concepts of classical Nevanlinna theory,as well as the foundation of value distribution theory in higher dimensions.Chapter 2 primarily investigates the value distribution results of holomorphic curves under more generalized conditions.Specifically,it combines Quang’s definition of distribution constants with the methodologies of Levin and Heier for handling closed subschemes.This approach yields the second main theorem and corresponding Schmidt subspace theorem for divisors with distribution constant positions and closed subschemes with the k-index in the lth general position.Compared to previous works,this chapter broadens the scope of position conditions and intersection targets,representing significant generalizations of earlier findings.Chapter 3 focuses on the value distribution problem of meromorphic mappings from pparabolic manifolds intersecting hypersurfaces with fixed distribution constant positions to projective algebraic varieties.It extends the research conducted by Q.Han,as well as W.Chen and Thin,and establishes the second main theorem and defect relation for meromorphic mappings from p-parabolic manifolds to projective algebraic varieties with truncated multiplicities.Chapter 4 introduces the study of the value distribution problem of meromorphic mappings from p-parabolic manifolds intersecting closed subschemes at arbitrary positions to projective algebraic varieties.In contrast to the approach of replacing hypersurfaces with Quang’s method,this chapter employs the generalized Chebyshev inequality in algebraic geometry to generalize the work under distribution constant conditions.Consequently,a second main theorem is established in the form of a weighted sum.Chapter 5 builds upon the existing definition of weighted height in weighted projective spaces and combines the Weil tools proposed by Salami and Shaska.This chapter defines weighted hypersurfaces and introduces new weighted standard metrics.It further establishes definitions for characteristic functions,counting functions,and approximation functions within weighted projective spaces.Finally,the main fundamental theorem for holomorphic curves of weighted hypersurfaces in the case of intersections at general positions is proven.Chapter 6 applies the second main theorem in weighted projective spaces,derived from Chapter 5,to proof refinement for holomorphic functions satisfying a diagonal equation on weighted projective spaces.This chapter examines the value distribution problem of holomorphic curves in the case of weighted hypersurfaces intersecting in a moving manner.Chapter 7 presents a comprehensive summary of the research pertaining to the value distribution problem of holomorphic curves.It emphasizes the innovative aspects of the research and offers insights into future research directions.