Open Access
August 2014 Curvatures of direct image sheaves of vector bundles and applications
Kefeng Liu, Xiaokui Yang
J. Differential Geom. 98(1): 117-145 (August 2014). DOI: 10.4310/jdg/1406137696


Let $\mathcal{p : X \to S}$ be a proper Kähler fibration and $\mathcal{E \to X}$ a Hermitian holomorphic vector bundle. As motivated by the work of Berndtsson (Curvature of vector bundles associated to holomorphic fibrations), by using basic Hodge theory, we derive several general curvature formulas for the direct image $\mathcal{p_* (K_{X/S} \otimes E)}$ for general Hermitian holomorphic vector bundle $\mathcal{E}$ in a simple way. A straightforward application is that, if the family $\mathcal{X \to S}$ is infinitesimally trivial and Hermitian vector bundle $\mathcal{E}$ is Nakano-negative along the base $\mathcal{S}$, then the direct image $\mathcal{p_* (K_{X/S} \otimes E)}$ is Nakano-negative. We also use these curvature formulas to study the moduli space of projectively flat vector bundles with positive first Chern classes and obtain that, if the Chern curvature of direct image $p_*(K_X \otimes E)$ —of a positive projectively flat family $(E, h(t))_{t \in \mathbb{D}} \to X$ —vanishes, then the curvature forms of this family are connected by holomorphic automorphisms of the pair $(X,E)$.


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Kefeng Liu. Xiaokui Yang. "Curvatures of direct image sheaves of vector bundles and applications." J. Differential Geom. 98 (1) 117 - 145, August 2014.


Published: August 2014
First available in Project Euclid: 23 July 2014

zbMATH: 1295.53066
MathSciNet: MR3263516
Digital Object Identifier: 10.4310/jdg/1406137696

Rights: Copyright © 2014 Lehigh University

Vol.98 • No. 1 • August 2014
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