Twisted Hodge filtration: Curvature of the determinant

TL;DR

This paper investigates the curvature properties of the determinant line bundle associated with the twisted Hodge filtration in holomorphic families of compact complex manifolds, establishing positivity results based on the positivity of the line bundle.

Contribution

It provides an explicit curvature formula for the determinant of the twisted Hodge filtration, demonstrating its (semi-) positivity under certain positivity conditions of the line bundle.

Findings

The curvature tensor of the direct images is explicitly computed.

The determinant of the twisted Hodge filtration is shown to be (semi-) positive.

Positivity depends on the (semi-) positivity of the line bundle L.

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 $R^{n-p}f_*\Omega^p_{\mathcal{X}/S}(L)$ 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 $\mathcal{X}$.