Variation of singular Kähler–Einstein metrics: Kodaira dimension zero (with an appendix by Valentino Tosatti)

Abstract

We study several questions involving relative Ricci-flat Kähler metrics for families of log Calabi-Yau manifolds. Our main result states that if $p:(X,B)\to Y$ is a Kähler fiber space such that $(X_y,B{|}_{X_y})$ is generically klt, $K_{X/Y}+B$ is relatively trivial and $p_{\ast }(m(K_{X/Y}+B))$ is Hermitian flat for some suitable integer $m$, then $p$ is locally trivial. Motivated by questions in birational geometry, we investigate the regularity of the relative singular Ricci-flat Kähler metric corresponding to a family $p:(X,B)\to Y$ of klt pairs $(X_y,B_y)$ such that $\kappa (K_{X_y}+B_y)=0$. Finally, we disprove a folkore conjecture by exhibiting a one-dimensional family of elliptic curves whose relative (Ricci-)flat metric is not semipositive.