On the Log Abundance for Compact {K{\"a}hler} threefolds II

TL;DR

This paper proves that for certain log canonical compact Kähler threefold pairs with a nef canonical divisor of numerical dimension two, the divisor is semi-ample, advancing the log abundance conjecture in this setting.

Contribution

It establishes the semi-ampleness of the canonical divisor for log canonical compact Kähler threefold pairs with numerical dimension two, completing the log abundance in this context.

Findings

Proves semi-ampleness of K_X + Δ under specified conditions.

Completes the log abundance conjecture for this class of threefolds.

Builds on previous work to confirm the conjecture in the Kähler setting.

Abstract

In this article we show that if $(X, \Delta)$ is a log canonical compact K\"ahler threefold pair such that $K_X+\Delta$ is nef and the numerical dimension $\nu(X, K_X+\Delta)=2$, then $K_X+\Delta$ is semi-ample. This result combined with our previous work in arXiv:2201.01202 shows that the log abundance holds for log canonical compact K\"ahler threefold pairs.