The structure of surfaces mapping to the moduli stack of canonically polarized varieties
@article{Kebekus2007TheSO, title={The structure of surfaces mapping to the moduli stack of canonically polarized varieties}, author={Stefan Kebekus and S{\'a}ndor Kov{\'a}cs}, journal={arXiv: Algebraic Geometry}, year={2007} }
Generalizing the well-known Shafarevich hyperbolicity conjecture, it has been conjectured by Viehweg that a quasi-projective manifold that admits a generically finite morphism to the moduli stack of canonically polarized varieties is necessarily of log general type. Given a quasi-projective surface that maps to the moduli stack, we employ extension properties of logarithmic pluri-forms to establish a strong relationship between the moduli map and the minimal model program of the surface. As a…
12 Citations
Birational geometry of smooth families of varieties admitting good minimal models
- Mathematics
- 2020
In this paper we study families of projective manifold with good minimal models. After constructing a suitable moduli functor for polarized varieties with canonical singularities, we show that, if…
Extension theorems for differential forms and Bogomolov–Sommese vanishing on log canonical varieties
- MathematicsCompositio Mathematica
- 2009
Abstract Given a normal variety Z, a p-form σ defined on the smooth locus of Z and a resolution of singularities $\pi : \widetilde {Z} \to Z$, we study the problem of extending the pull-back π*(σ)…
Brody hyperbolicity of base spaces of certain families of varieties
- Mathematics
- 2018
We prove that quasi-projective base spaces of smooth families of minimal varieties of general type with maximal variation do not admit Zariski dense entire curves. We deduce the fact that moduli…
Generic positivity and applications to hyperbolicity of moduli spaces
- Mathematics
- 2016
The proof of the celebrated Viehweg's hyperbolicity conjecture is a consequence of two remarkable results: Viehweg and Zuo's existence results for global pluri-differential forms induced by variation…
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…
Differential forms on singular spaces, the minimal model program, and hyperbolicity of moduli stacks
- Mathematics
- 2011
This survey discusses hyperbolicity properties of moduli stacks and generalisations of the Shafarevich Hyperbolicity Conjecture to higher dimensions. It concentrates on methods and results that…
Differential forms in positive characteristic avoiding resolution of singularities
- Mathematics
- 2014
This paper studies several notions of sheaves of differential forms that are better behaved on singular varieties than K\"ahler differentials. Our main focus lies on varieties that are defined over…
References
SHOWING 1-10 OF 28 REFERENCES
Families of canonically polarized varieties over surfaces
- Mathematics
- 2005
Shafarevich’s hyperbolicity conjecture asserts that a family of curves over a quasi-projective 1-dimensional base is isotrivial unless the logarithmic Kodaira dimension of the base is positive. More…
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…
The geometry of moduli spaces of sheaves
- Mathematics
- 1997
Preface to the second edition Preface to the first edition Introduction Part I. General Theory: 1. Preliminaries 2. Families of sheaves 3. The Grauert-Mullich Theorem 4. Moduli spaces Part II.…
Rationally connected foliations after Bogomolov and McQuillan
- Mathematics
- 2005
This paper is concerned with a sufficient criterion to guarantee that a given foliation on a normal variety has algebraic and rationally connected leaves. Following ideas from a preprint of…
Existence of Rational Curves on Algebraic Varieties, Minimal Rational Tangents, and Applications
- Mathematics
- 2006
This survey paper discusses some of the recent progress in the study of rational curves on algebraic varieties. It was written for the survey volume of the priority programme "Global Methods in…
Positivity Of Direct Image Sheaves And Applications To Families Of Higher Dimensional Manifolds
- Mathematics
- 2000
L. Caporaso [4] has shown recently, that the number of non-isotrivial families in (I) is bounded by a constant depending only on the genus q of the fibre, on the genus g of Y and on s = #S.…
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…
Birational Geometry of Algebraic Varieties
- Mathematics
- 1998
1. Rational curves and the canonical class 2. Introduction to minimal model program 3. Cone theorems 4. Surface singularities 5. Singularities of the minimal model program 6. Three dimensional flops…
A General Non-Vanishing Theorem and an Analytic Proof of the Finite Generation of the Canonical Ring
- Mathematics
- 2006
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…
Rational curves on quasi-projective surfaces
- Mathematics
- 1997
Introduction and statement of results Glossary of notation and conventions Gorenstein del Pezzo surfaces Bug-eyed covers Log deformation theory Criteria for log uniruledness Reduction to…