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

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 $K_X$.