Algebraic degeneracy of holomorphic curves

TL;DR

This paper investigates the algebraic degeneracy of holomorphic curves using meromorphic vector fields and Jacobian sections, establishing a Second Main Theorem type inequality and deriving new degeneracy results for curves in projective varieties.

Contribution

It introduces a novel approach employing meromorphic vector fields and Jacobian sections to prove a Second Main Theorem type inequality for holomorphic curves.

Findings

Established a Second Main Theorem type inequality for holomorphic curves.

Derived several algebraic degeneracy theorems for curves in complex projective varieties.

Provided a new perspective on the algebraic degeneracy using meromorphic vector fields.

Abstract

We consider the algebraic degeneracy of holomorphic curves from a point of view of meromorphic vector fields. Employing the notion of Jocabian sections introduced by W. Stoll, we establish a Second Main Theorem type inequality. As consequences, several algebraic degeneracy theorems are obtained for holomorphic curves into a complex projective variety.