Positivity of direct images with a Poincaré type twist

We consider a holomorphic family \(f:\mathcal{X} \to S\) of compact complex manifolds and a line bundle \(\mathcal{L}\to \mathcal{X}\). Given that \(\mathcal{L}^{-1}\) carries a singular hermitian metric that has Poincaré type singularities along a relative snc divisor \(\mathcal{D}\), the direct image \(f_*(K_{\mathcal{X}/S}\otimes \mathcal{D} \otimes \mathcal{L})\) carries a smooth hermitian metric. In case \(\mathcal{L}\) is relatively positive, we give an explicit formula for its curvature. The result applies to families of log-canonically polarized pairs. Moreover we show that it improves the general positivity result of Berndtsson-Păun in a special situation of a big line bundle.