Curvature of higher direct images

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 \(f_*(K_{\mathcal{X}/F}\otimes L)\).