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

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.