We study several questions involving relative Ricci-flat Kahler metrics for families of log Calabi-Yau manifolds. Our main result states that if $p:(X,B)\to Y$ is a Kahler fiber space such that $\displaystyle (X_y, B|_{X_y})$ is generically klt, $K_{X/Y}+B$ is relatively trivial and $p_*(m(K_{X/Y}+B))$ 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 Kahler 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.

中文翻译：

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奇异 Kähler-Einstein 度量的变化：Kodaira 维数为零

我们研究了几个涉及对数 Calabi-Yau 流形族的相对 Ricci-flat Kahler 度量的问题。我们的主要结果表明，如果 $p:(X,B)\to Y$ 是一个 Kahler 纤维空间，使得 $\displaystyle (X_y, B|_{X_y})$ 一般是 klt, $K_{X/Y} +B$ 是相对平凡的，$p_*(m(K_{X/Y}+B))$ 是某个合适整数 $m$ 的 Hermitian flat，那么 $p$ 是局部平凡的。受双有理几何问题的启发，我们研究了对应于 klt 对 $(X_y,B_y)$ 的族 $p:(X,B)\to Y$ 的相对奇异 Ricci-flat Kahler 度量的规律性，使得 $\卡帕（K_{X_y}+B_y）=0$。最后，我们通过展示其相对（Ricci-）平坦度量不是半正的一维椭圆曲线族来反驳民间猜想。