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 $(X,E)$.