Variation of singular Kähler–Einstein metrics: Kodaira dimension zero

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 fiber space such $(X\_y, B|{X\_y})$ generically klt, $K{X/Y}+B$ relatively trivial and $p\_\*(m(K\_{X/Y}+B))$ Hermitian flat some suitable integer $m$, then $p$ locally trivial. Motivated by in birational geometry, we investigate the regularity singular metric corresponding to family klt pairs $(X\_y,B\_y)$ $\kappa(K\_{X\_y}+B\_y)=0$. Finally, disprove folkore conjecture exhibiting one-dimensional elliptic curves whose (Ricci-)flat not semipositive.