Curvatures of direct image sheaves of vector bundles and applications I

Let $p:\text{sXS}$ be a proper K"ahler fibration and $\text{sE}\text{sX}$ a Hermitian holomorphic vector bundle. As motivated by the work of Berndtsson(\cite{Berndtsson09a}), by using basic Hodge theory, we derive several general curvature formulas for the direct image $p_{\ast }(K_{\text{sX}\mathrm{/}S}\text{ts}\text{sE})$ for general Hermitian holomorphic vector bundle $\text{sE}$ in a simple way. A straightforward application is that, if the family $\text{sXS}$ is infinitesimally trivial and Hermitian vector bundle $\text{sE}$ is Nakano-negative along the base $S$, then the direct image $p_{\ast }(K_{\text{sX}\mathrm{/}S}\text{ts}\text{sE})$ 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 $p_{\ast }(K_X\text{ts}E)$–of a positive projectively flat family $(E,h(t){)}_{t\in \mathbb{D}}X$–vanishes, then the curvature forms of this family are connected by holomorphic automorphisms of the pair $(X,E)$.