Let $p:X\rightarrow Y$ be an algebraic fiber space, and let $L$ be a line bundle on $X$ . In this article, we obtain a curvature formula for the higher direct images of $\unicode[STIX]{x1D6FA}_{X/Y}^{i}\otimes L$ restricted to a suitable Zariski open subset of $X$ . Our results are particularly meaningful if $L$ is semi-negatively curved on $X$ and strictly negative or trivial on smooth fibers of $p$ . Several applications are obtained, including a new proof of a result by Viehweg–Zuo in the context of a canonically polarized family of maximal variation and its version for Calabi–Yau families. The main feature of our approach is that the general curvature formulas we obtain allow us to bypass the use of ramified covers – and the complications that are induced by them.

中文翻译：

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代数光纤空间和更高直接图像的曲率

让 $p:X\右箭头 Y$ 是代数纤维空间，令 $L$ 成为线束 $X$ . 在本文中，我们获得了更高直接图像的曲率公式 $\unicode[STIX]{x1D6FA}_{X/Y}^{i}\otimes L$ 限于合适的 Zariski 开子集 $X$ . 我们的结果特别有意义，如果 $L$ 是半负弯曲的 $X$ 并且在光滑的纤维上严格为负或微不足道 $p$ . 获得了几个应用程序，包括 Viehweg-Zuo 在最大变异的规范极化家族及其对 Calabi-Yau 家族的版本的背景下对结果的新证明。我们方法的主要特点是，我们获得的一般曲率公式允许我们绕过分枝覆盖的使用——以及由它们引起的复杂性。