Twisted Hodge filtration: Curvature of the determinant

Given a holomorphic family f:X→S of compact complex manifolds and a relatively ample line bundle L→X, the higher direct images Rn−pf∗ΩX/Sp(L) carry a natural hermitian metric. An explicit formula for the curvature tensor of these direct images is given in [8]. We prove that the determinant of the twisted Hodge filtration FLp=⨁i≥pRn−if∗ΩX/Si(L) is (semi‐)positive on the base S if L itself is (semi‐)positive on X.