Existence of cscK metrics on smooth minimal models

Abstract

Given a compact Kähler manifold $X$ it is interesting to ask whether it admits a constant scalar curvature Kähler (cscK) metric. In this short note we show that there always exist cscK metrics on compact Kähler 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 \cite{JianShiSong} 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. As a byproduct we obtain that the connected component ${\mathrm{Aut}}_0(X)$ of a compact Kähler manifold with $K_X$ nef is either trivial or a complex torus.

9 pages, new result added (Corollary 2). Accepted in Annali Della Scuola Normale Superiore Di Pisa - Classe Di Scienze