Compactifications of smooth families and of moduli spaces of polarized manifolds
type: Item
instance of: scholarly article
title: Compactifications of smooth families and of moduli spaces of polarized manifolds
zbMATH Open document ID: 1238.14009
author: Eckart Viehweg
published in: Annals of Mathematics. Second Series
publication date: 2010-10-29T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0605093
review text: Let \(M_{h}\) be the moduli scheme of canonically polarized manifolds with Hilbert polynomial \(h\). One constructs in this paper for \(\nu\geq2\) with \(h(\nu)>0\) a projective compactification \(\overline{M}_{h}\) of the reduced moduli scheme \((M_{h})_{\mathrm{red}}\) such that the ample invertible sheaf \(\lambda_{\nu}\), corresponding to \(\det(f_{*}\omega_{X_{0}/Y_{0}}^{\nu})\) on the moduli stack, has a natural extension \(\overline{\lambda}_{\nu}\in\text{Pic}(\overline{M}_{h})_{\mathbb{Q}}\). A similar result is shown for moduli of polarized minimal models of Kodaira dimension zero. The useful technical statements are: Theorem 1. Let \(f_{0}:X_{0}\rightarrow Y_{0}\) be a smooth projective morphism of quasi-projective reduced schemes such that \(\omega_{F}\) is semiample for all fibers \(F\) of \(f_{0}\). Let \(I\) be a finite set of positive integers. Then there exists a projective compactification \(Y\) of \(Y_{0}\), a finite covering \(\phi:W\rightarrow Y\) with a splitting trace map, and for \(\nu\in I\) a locally free sheaf \(\mathcal{F}_{W}^{(\nu)}\) on W with: (i) For \(W_{0}=\phi^{-1}(Y_{0})\) and \(\phi_{0}=\phi|_{W_{0}}\) one has \(\phi_{0}^{*}f_{0*}\omega^{\nu}_{X_{0}/Y_{0}}=\mathcal{F}_{W}^{(\nu)}|_{W_{0}}\). (ii) Let \(\xi:\widehat{Y}\rightarrow W\) be a morphism from a nonsingular projective variety \(\widehat{Y}\) with \(\widehat{Y}_{0}=\xi^{-1}(W_{0})\) dense in \(\widehat{Y}\). Assume either that \(\widehat{Y}\) is a curve, or that \(\widehat{Y}\rightarrow W\) is dominant. For some \(r\geq1\) let \(X^{(r)}\) be a nonsingular projective model of the \(r\)-fold product family \(\widehat{X}_{0}^{r}=(X_{0}\times_{Y_{0}}\times \dots \times_{Y_{0}}X_{0})\times_{Y_{0}}\widehat{Y}_{0}\), which admits a morphism \(f^{(r)}:X^{(r)}\rightarrow\widehat{Y}\). Then \(f_{*}^{(r)}\omega^{\nu}_{X^{(r)}/\widehat{Y}}=\bigotimes^{r}\xi^{*}\mathcal{F}_{W}^{(\nu)}\). The second result is Theorem 2. Conditions (i) and (ii) in Theorem 1 imply: (iii) The sheaf \(\math…
Mathematics Subject Classification ID: 14D20
Mathematics Subject Classification ID: 14J10
Mathematics Subject Classification ID: 14J40
Mathematics Subject Classification ID: 14F18
zbMATH DE Number: 5808578
zbMATH Keywords: moduli schemes
zbMATH Keywords: compactification
zbMATH Keywords: weak semistable reduction
zbMATH Keywords: multiplier ideals
reviewed by: Vasile Brînzănescu
MaRDI profile type: Publication
OpenAlex ID: W2089287407
arXiv ID: math/0605093
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DOI: 10.4007/ANNALS.2010.172.809
description: scientific article