ALGEBRAIC FIBER SPACES AND CURVATURE OF HIGHER DIRECT IMAGES. (3rd May 2022)

• Record Type: Journal Article
• Authors: Berndtsson, Bo; Păun, Mihai; Wang, Xu
• Is Part Of: Journal of the Institute of Mathematics of Jussieu. Volume 21:Number 3(2022)
• Page Start: 973
• Page End: 1028
• Publication Date: 2022-05-03
• Subjects: higher direct imags of relative canonical bundle -- Hodge theory -- posititivity -- curvture -- singular metrics; 53Cxx -- 58Axx -- 32A70; Mathematics -- Periodicals; 510.5
• DOI: 10.1017/S147474802000050X
• Languages: English
• ISSNs: 1474-7480

Abstract

Abstract: 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.