Curvatures of Direct Image Sheaves of Vector Bundles

Let p : X → S be a smooth Kähler fibration and E → X a Hermitian holomorphic vector bundle. As motivated by the work of Berdtsson([Bern09]), by using basic Hodge theory, we derive several general curvature formulas for the direct image p∗(KX/S ⊗E) for general Hermitian holomorphic vector bundle E in a very simple way. A straightforward application is that, if the Hermitian vector bundle E is Nakano-negative along the base S, then the direct image p∗(KX/S ⊗ 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∗(KX ⊗ E)–of a positive projectively flat family (E, h(t))t∈D → X–vanishes, then the curvature forms of this family are connected by holomorphic automorphisms of the pair (X,E).