Abundance for semi log canonical compact Kähler threefolds

Abstract

In this article we show that if $(X,\Delta )$ is a compact Kähler slc pair of dimension $3$ such that $K_X+\Delta$ is nef, then $K_X+\Delta$ is semiample. Moreover, we extend our result to a special subclass of Fujiki class $\mathcal{C}$ varieties, and show that in this subclass, if $(X,\Delta )$ is a slc pair and $K_X+\Delta$ is nef but not numerically trivial, then it is semiample. Moreover, we also construct a slc pair $(X,\Delta )$ in Fujiki class $\mathcal{C}$ such that $K_X+\Delta \equiv 0$ but $K_X+\Delta$ is not semiample. In particular, the slc abundance fails for Fujiki class $\mathcal{C}$ varieties.

arXiv commentMajor revision. The sdlt abundance result remains valid, with some arguments modified. The previously claimed counterexample is not valid in the compact Kähler setting; the same construction gives counterexample in the Fujiki class. Abundance is now proved in full generality for compact Kähler slc pairs. Comments are welcome