  Given a holomorphic family $f:\mathcal{X} \to S$ of compact complex manifolds
and a relative 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. Using
the explicit formula for the curvature tensor of these direct images, we prove
that the determinant of the twisted Hodge filtration $F^p_L=\oplus_{i\geq
p}R^{n-i}\Omega^i_{\mathcal{X}/S}(L)$ is (semi-) positive on the base $S$ if
$L$ itself is (semi-) positive on $\mathcal{X}$.
