Positivity of relative canonical bundles for families of canonically polarized manifolds

TL;DR

This paper proves that the relative canonical bundle of a family of canonically polarized manifolds admits a strictly positive metric, providing an analytic approach to the quasi-projectivity of their moduli space.

Contribution

It introduces a new analytic method using elliptic equations to establish positivity of the relative canonical bundle in families of polarized manifolds.

Findings

The induced metric on the relative canonical bundle is strictly positive for effectively parameterized families.

A singular hermitian metric is constructed for degenerating families.

The approach offers an analytic proof of the quasi-projectivity of the moduli space.

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.