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

Supercedes arXiv:0808.3259v4 and arXiv:1002.4858v2. To appear in Invent. math

References in corpus (2)

• Log canonical singularities are Du Bois
• Hodge metrics and the curvature of higher direct images

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• Second variation of Selberg zeta functions and curvature asymptotics
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