Positivity of relative canonical bundles and applications

Abstract

Given a family \(f:\mathcal{X} \to S\) of canonically polarized manifolds, the unique Kähler–Einstein metrics on the fibers induce a hermitian metric on the relative canonical bundle \(\mathcal{K}_{\mathcal{X}/S}\). We use a global elliptic equation to show that this metric is strictly positive on \(\mathcal{X}\), unless the family is infinitesimally trivial.

For degenerating families we show that the curvature form on the total space can be extended as a (semi-)positive closed current. By fiber integration it follows that the generalized Weil–Petersson form on the base possesses an extension as a positive current. We prove an extension theorem for hermitian line bundles, whose curvature forms have this property. This theorem can be applied to a determinant line bundle associated to the relative canonical bundle on the total space. As an application the quasi-projectivity of the moduli space \(\mathcal{M}_{\text{can}}\) of canonically polarized varieties follows.

The direct images \(R^{n-p}f_{*}\Omega^{p}_{\mathcal{X}/S}(\mathcal {K}_{\mathcal{X}/S}^{\otimes m})\), m>0, carry natural hermitian metrics. We prove an explicit formula for the curvature tensor of these direct images. We apply it to the morphisms \(S^{p}\mathcal{T}_{S} \to R^{p}f_{*}\Lambda^{p}\mathcal{T}_{\mathcal{X}/S}\) that are induced by the Kodaira–Spencer map and obtain a differential geometric proof for hyperbolicity properties of \(\mathcal{M}_{\text{can}}\).