# Compactifications of smooth families and of moduli spaces of polarized manifolds

@article{Viehweg2006CompactificationsOS,
title={Compactifications of smooth families and of moduli spaces of polarized manifolds},
author={Eckart Viehweg},
journal={Annals of Mathematics},
year={2006},
volume={172},
pages={809-910}
}
• E. Viehweg
• Published 3 May 2006
• Mathematics
• Annals of Mathematics
Let M h be the moduli scheme of canonically polarized manifolds with Hilbert polynomial h. We construct for v ≥ 2 with h(v) > 0 a projective compactification M h of the reduced moduli scheme (M h ) red such that the ample invertible sheaf λ ν , corresponding to det( f * ω v X0/Y0 ) on the moduli stack, has a natural extension λ ν ∈ Pic(M h ) ℚ . A similar result is shown for moduli of polarized minimal models of Kodaira dimension zero. In both cases "natural" means that the pullback of λ ν to a…
21 Citations
Arakelov Inequalities
The proof of the Shafarevich Conjecture for curves of genus g ≥ 2 over complex function fields K = C(Y ), given by Arakelov in [AR71], consists of two parts, the verification of “boundedness” and of
Topological methods in moduli theory
One of the main themes of this long article is the study of projective varieties which are K(H,1)’s, i.e. classifying spaces BH for some discrete group H. After recalling the basic properties of such
Special families of curves, of Abelian varieties, and of certain minimal manifolds over curves
• Mathematics
• 2006
This survey article discusses some results on the structure of families f:V-->U of n-dimensional manifolds over quasi-projective curves U, with semistable reduction over a compactification Y of U. We
Ja n 20 12 POSITIVITY OF RELATIVE CANONICAL BUNDLES AND APPLICATIONS
Given a family f : X → S of canonically polarized manifolds, the unique Kähler-Einstein metrics on the fibers induce a hermitian metric on the relative canonical bundle KX/S. We use a global elliptic
The KSBA compactification for the moduli space of degree two K3 pairs
Inspired by the ideas of the minimal model program, Shepherd-Barron, Koll\'ar, and Alexeev have constructed a geometric compactification for the moduli space of surfaces of log general type. In this
EFFECTIVE IITAKA FIBRATIONS ECKART VIEHWEG AND
For every n-dimensional projective manifold X of Kodaira dimension 2 we show that Φ|MKX | is birational to an Iitaka fibration for a computable positive integer M = M(b, Bn−2), where b > 0 is minimal
Positivity of relative canonical bundles and applications
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
A Superficial Working Guide to Deformations and Moduli
This is the first part of a guide to deformations and moduli, especially viewed from the perspective of algebraic surfaces (the simplest higher dimensional varieties). It contains also new results,
Differential forms on log canonical spaces
• Mathematics
• 2010
The present paper is concerned with differential forms on log canonical varieties. It is shown that any p-form defined on the smooth locus of a variety with canonical or klt singularities extends
A Hodge theoretic criterion for finite Weil--Petersson degenerations over a higher dimensional base
We give a Hodge-theoretic criterion for a Calabi--Yau variety to have finite Weil--Petersson distance on higher dimensional bases up to a set of codimension $\geq 2$. The main tool is variation of

## References

SHOWING 1-10 OF 27 REFERENCES
Base Spaces of Non-Isotrivial Families of Smooth Minimal Models
• Mathematics
• 2002
Given a polynomial h of degree n let M h be the moduli functor of canonically polarized complex manifolds with Hilbert polynomial h. By [23] there exist a quasi-projective scheme M h together with a
Quasi-projective moduli for polarized manifolds
• E. Viehweg
• Mathematics
Ergebnisse der Mathematik und ihrer Grenzgebiete
• 1995
This text discusses two subjects of quite different natures: construction methods for quotients of quasi-projective schemes either by group actions or by equivalence relations; and properties of
Weak semistable reduction in characteristic 0
• Mathematics
• 1997
Let X->B be a morphism of varieties in characteristic zero. Semistable reduction has been proved for dim(B)=1 (Kempf, Knudsen, Mumford, Saint-Donat), dim(X)=dim(B)-1 (de Jong) and dim(X)=dim(B)+2
On the Brody hyperbolicity of moduli spaces for canonically polarized manifolds
• Mathematics
• 2000
We show that the moduli stack Mh of canonically polarized complex manifolds with Hilbert polynomial h is Brody hyperbolic. Hence if Mh denotes the corresponding coarse moduli scheme, and if U → Mh is
Minimal models and boundedness of stable varieties
We consider a class of stable smoothable n-dimensional varieties, the analogs of stable curves. Assuming the minimal model program in dimension n+1, we prove that this class is bounded. From Kollar's
Erratum for Boundedness of families of canonically polarized manifolds: A higher dimensional analogue of Shafarevich's conjecture
• Mathematics
• 2006
We show that the number of deformation types of canonically polarized manifolds over an arbitrary variety with proper singular locus is nite, and that this number is uniformly bounded in any nite
Lectures on Vanishing Theorems
• Mathematics
• 2004
1 Kodaira's vanishing theorem, a general discussion.- 2 Logarithmic de Rham complexes.- 3 Integral parts of Q-divisors and coverings.- 4 Vanishing theorems, the formal set-up.- 5 Vanishing theorems
Geometric Invariant Theory
“Geometric Invariant Theory” by Mumford/Fogarty (the first edition was published in 1965, a second, enlarged edition appeared in 1982) is the standard reference on applications of invariant theory to
Compactifying the space of stable maps
• Mathematics
• 1999
We define two equivalent notions of twisted stable map from a curve to a Deligne-Mumford stack with projective moduli space, and we prove that twisted stable maps of fixed degree form a complete
A General Non-Vanishing Theorem and an Analytic Proof of the Finite Generation of the Canonical Ring
On August 5, 2005 in the American Mathematical Society Summer Institute on Algebraic Geometry in Seattle and later in several conferences I gave lectures on my analytic proof of the finite generation