Positivity of relative canonical bundles for families of canonically polarized manifolds

Abstract

Given an effectively parameterized family of canonically polarized manifolds the Kaehler-Einstein metrics on the fibers induce a hermitian metric on the relative canonical bundle. We use a global elliptic equation to show that this metric is strictly positive. For degenerating families we obtain a singular hermitian metric. Main application is an analytic proof of the quasi-projectivity of the moduli space of canonically polarized manifolds. Further applications in arXiv:1002.4858v2.

Details added

References in corpus (2)

• Hodge metrics and the curvature of higher direct images
• Compactifications of smooth families and of moduli spaces of polarized manifolds

Cited by in corpus (9)

• Kahler-Einstein metrics emerging from free fermions and statistical mechanics
• Understanding Weil-Petersson curvature
• Strict and non strict positivity of direct image bundles
• Moduli of canonically polarized manifolds, higher order Kodaira-Spencer maps, and an analogy to Calabi-Yau manifolds
• On the hyperbolicity of base spaces for maximally variational families of smooth projective varieties
• Variation of singular Kähler-Einstein metrics: positive Kodaira dimension
• Variation of geodesic length functions over Teichmüller space
• Holomorphic curves in Base Spaces of Families of Polarized Manifolds
• Subharmonicity of the variations of Kähler-Einstein metrics on pseudoconvex domains