Existence of cscK metrics on smooth minimal models

Abstract

Given a compact Kahler manifold $X$ it is interesting to ask whether it admits a constant scalar curvature Kahler (cscK) metric in at least one Kahler class $[\omega] \in H^{1,1}(X,\mathbb{R})$. In this short note we show that there always exist cscK metrics on compact Kahler manifolds with nef canonical bundle, thus on all smooth minimal models, and also on the blowup of any such manifold. This confirms an expectation of Jian-Shi-Song (arXiv:1805.06863) and extends their main result from $K_X$ semi-ample to $K_X$ nef, with a direct proof that does not appeal to the Abundance conjecture.