An approach to the abundance conjecture for K\"ahler varieties via algebraic reduction

TL;DR

This paper proposes a new inductive approach to the abundance conjecture for K"ahler varieties by reducing it to the projective case through algebraic reduction, with applications to fourfolds.

Contribution

It introduces a strategy to prove the abundance conjecture for K"ahler varieties by induction on algebraic dimension, extending results to certain fourfolds.

Findings

Reduced the abundance conjecture for K"ahler varieties to the projective case.

Applied the strategy to obtain results for non-algebraic fourfolds.

Provided a framework for inductive proofs in complex geometry.

Abstract

In this article, we establish a strategy to the abundance conjecture for K\"ahler varieties via induction on algebraic dimension. Our strategy is to reduce the abundance conjecture for K\"ahler varieties to the abundance conjecture for projective varieties using the algebraic reduction fibration. In dimension 4, we apply our inductive strategy to obtain some cases of the abundance conjecture for K\"ahler fourfolds that are not algebraic or have trivial $K_X$.