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Published online by Cambridge University Press:  03 September 2020

Bo Berndtsson
Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, SE-412 96Gothenburg, Sweden (
Mihai Păun
Mathematisches Institut der Universität Bayreuth, Germany (
Xu Wang
Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491Trondheim, Norway (


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.

MSC classification

Research Article
© The Author(s) 2020. Published by Cambridge University Press

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