Failure of the semi log canonical Abundance for compact K\"{a}hler threefolds

TL;DR

This paper demonstrates the failure of semi log canonical abundance in compact K"ahler threefolds through a counterexample, but also establishes conditions under which abundance holds for certain pairs.

Contribution

It constructs a counterexample showing failure of semi log canonical abundance in dimension 3, and proves semi-ampleness under specific conditions for K"ahler pairs.

Findings

Counterexample of a compact K"ahler slc threefold with nef but not semiample canonical bundle.

Semi-ampleness holds for compact K"ahler semi-dlt pairs with nef canonical bundle.

Semi-ampleness also holds for slc pairs with positive Kodaira dimension on all components.

Abstract

In this article we show that the semi log canonical abundance for compact K\"ahler varieties fails in dimension $3$. More specifically we construct a counterexample of a compact K\"ahler (irreducible) slc threefold $(X, 0)$ such that $K_X$ is nef and $\kappa(\tilde X, K_{\tilde X}+\tilde D)=0$, where $\mu:(\tilde X, \tilde D)\to X$ is the normalization morphism, but $K_X$ is not semiample. On the other hand, we show that if we start with a compact K\"ahler semi-dlt pair, then the abundance does hold, i.e., if $(X, \Delta)$ is a compact K\"ahler sdlt pair of dimension $3$ such that $K_X+\Delta$ is nef, then it is semiample. We also show that if $(X, \Delta)$ is a compact K\"ahler slc pair of dimension $3$, $K_X+\Delta$ is nef, and $\kappa(X'_i, \Delta'_i+D'_i)>0$ for all $i$, where $\mu:\sqcup(X'_i, \Delta'_i+D'_i)\to (X,\Delta)$ is the normalization, then $K_X+\Delta$ is semiample.