On the Log Abundance for Compact K\"ahler $3$-folds

TL;DR

This paper proves that for certain log canonical compact K"ahler 3-folds with nef canonical divisors, the divisors are semi-ample unless their numerical dimension equals 2.

Contribution

It establishes semi-ampleness of the canonical divisor on log canonical compact K"ahler 3-folds under specific numerical conditions, extending previous results.

Findings

Semi-ampleness holds when the numerical dimension is not 2.

The result applies to log canonical compact K"ahler 3-folds with nef canonical divisors.

Provides conditions under which the canonical divisor is semi-ample.

Abstract

In this article we show that if $(X, \Delta)$ is a log canonical compact K\"ahler $3$-fold such that $K_X+\Delta$ is nef and the numerical dimension $\nu(K_X+\Delta)\neq 2$, then $K_X+\Delta$ is semi-ample.