Positivity of relative canonical bundles and applications

By G. Schumacher. 2012

113 citations recorded by the source corpus.

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 $\mathcal{M}_{\text{can}}$.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/25c5a3cd6c485a2f2fb82e8f2a7678ce282e82ed

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Kähler–Einstein Metrics on Stable Varieties and log Canonical Pairs (2013, background): “To be a bit more precise we ﬁrst recall that given a line bundle L → X over a (complex) n − dimensional algebraic variety X its top Deligne pairing i.e. the ( n +1) − fold Deligne pairing of L with itself is a complex line that we will denote by (cid:104) L (cid:105) [ Elk89, Elk90 ].”
Finsler metrics and Kobayashi hyperbolicity of the moduli spaces of canonically polarized manifolds (2015, background): “…this work, our attention was drawn to a recent preprint (arXiv 1002.4858 ) by Schumacher (which has appeared subsequently as [Sch12]), which, among other results, gives rise to Finsler metrics of negative holomorphic sectional curvature on relatively compact subsets of S (see [Sch12, Prop.”