Curvature of higher direct images

Abstract

Given a holomorphic family $f:\mathcal{X} \to S$ of compact complex manifolds of dimension $n$ and a relatively ample line bundle $L\to \mathcal{X}$, the higher direct images $R^{n-p}f_*\Omega^p_{\mathcal{X}/S}(L)$ carry a natural hermitian metric. We give an explicit formula for the curvature tensor of these direct images. This generalizes the result of Schumacher, where he computed the curvature of $R^{n-p}f_*\Omega^p_{\mathcal{X}/S}(K_{\mathcal{X}/S}^{\otimes m})$ for a family of canonically polarized manifolds. For $p=n$, it coincides with a formula of Berndtsson. Thus, when $L$ is globally ample, we reprove his result on the Nakano positivity of $f_*(K_{\mathcal{X}/F}\otimes L)$.