2012/01/13 by Georg Schumacher
Mathematics · #math.AG #math.CV #math.DG #msc:14D20 #msc:32L10 #msc:32Q20
paper · pdf · doi:10.1007/s00222-012-0374-7
published as Invent. Math. 190, No. 1, 1-56 (2012) and Invent. Math. 192, No. 1, 253-255 (2013) · Supercedes arXiv:0808.3259v4 and arXiv:1002.4858v2. To appear in Invent. math
arxiv created 2012/01/13 · arxiv updated 2015/06/03
Given a family f:\mathcal X → 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 Mcan of canonically polarized varieties follows. The direct images Rn-pf_*Ωp\mathcal X/S(\mathcal K\mathcal X/S⊗ 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 Sp \mathcal TS → Rpf_*Λp\mathcal T\mathcal X/S that are induced by the Kodaira-Spencer map and obtain a differential geometric proof for hyperbolicity properties of \mathcal Mcan.