## Journal of Differential Geometry

### Curvatures of direct image sheaves of vector bundles and applications

#### Abstract

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 $(X,E)$.

#### Article information

Source
J. Differential Geom., Volume 98, Number 1 (2014), 117-145.

Dates
First available in Project Euclid: 23 July 2014

https://projecteuclid.org/euclid.jdg/1406137696

Digital Object Identifier
doi:10.4310/jdg/1406137696

Mathematical Reviews number (MathSciNet)
MR3263516

Zentralblatt MATH identifier
1295.53066

#### Citation

Liu, Kefeng; Yang, Xiaokui. Curvatures of direct image sheaves of vector bundles and applications. J. Differential Geom. 98 (2014), no. 1, 117--145. doi:10.4310/jdg/1406137696. https://projecteuclid.org/euclid.jdg/1406137696