# Abundance conjecture

## TL;DR

This paper sketches a proof of the abundance conjecture, establishing the equality of Kodaira and numerical Kodaira dimensions for compact complex algebraic manifolds using advanced techniques from complex geometry and number theory.

## Contribution

It provides a new proof of the abundance conjecture by combining Simpson's method, semi-positivity, and Nevanlinna theory techniques, addressing key cases and assumptions.

## Key findings

- Proof of the abundance conjecture under certain conditions.
- Verification of the coincidence of foliations and fibrations for the canonical bundle.
- Application of Gelfond-Schneider and Nevanlinna theory techniques.

## Abstract

We sketch a proof of the abundance conjecture that the Kodaira dimension of a compact complex algebraic manifold equals its numerical Kodaira dimension. The proof consists of the following three parts: (i) the case of numerical Kodaira dimension zero, (ii) the general case under the assumption of the coincidence of the numerically trivial foliation and fibration for the canonical bundle, and (iii) the verification of the coincidence of the numerically trivial foliation and fibration for the canonical bundle. Besides the use of standard techniques such as the L2 estimates of d-bar, the first part uses Simpson's method of replacing the flat line bundle in a nontrivial flatly twisted canonical section by a torsion flat line bundle. Simpson's method relies on the technique of Gelfond-Schneider for the solution of the seventh problem of Hilbert. The second part uses the semi-positivity of the direct image of a relative pluricanonical bundle. The third part uses the technique of the First Main Theorem of Nevanlinna theory and its use is related to the technique of Gelfond-Schneider in the first part.