Twisted Hodge filtration: Curvature of the determinant

Abstract

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 $R^{n-p}{f}_{\ast }{\Omega }_{\mathcal{X}/S}^p(L)$ 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 $F_L^p={\oplus }_{i\ge p}{R}^{n-i}{\Omega }_{\mathcal{X}/S}^i(L)$ is (semi-) positive on the base $S$ if $L$ itself is (semi-) positive on $\mathcal{X}$.

arXiv commentarXiv admin note: substantial text overlap with arXiv:1611.09117