Differential Forms on Log Canonical Spaces
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In this paper, it was shown that any p-form defined on the smooth locus of a variety with canonical or klt singularities extends regularly to any resolution of singularities.Abstract:
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 regularly to any resolution of singularities. In fact, a much more general theorem for log canonical pairs is established. The proof relies on vanishing theorems for log canonical varieties and on methods of the minimal model program. In addition, a theory of differential forms on dlt pairs is developed. It is shown that many of the fundamental theorems and techniques known for sheaves of logarithmic differentials on smooth varieties also hold in the dlt setting. Immediate applications include the existence of a pull-back map for reflexive differentials, generalisations of Bogomolov-Sommese type vanishing results, and a positive answer to the Lipman-Zariski conjecture for klt spaces.read more
Citations
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Book ChapterDOI
Aspects of the Geometry of Varieties with Canonical Singularities
Stefan Kebekus,Thomas Peternell +1 more
TL;DR: A survey of recent developments regarding the global structure of complex varieties which occur in the minimal model program can be found in this paper, where the authors present a survey of the most recent developments.
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The structure of algebraic varieties
TL;DR: The main questions and results of the structure theory of higher dimensional algebraic varieties can be found in this paper, where the authors give an overview of the main results and main questions.
Journal ArticleDOI
On the Lipman-Zariski conjecture for logarithmic vector fields on log canonical pairs
TL;DR: In this article, a version of the Lipman-Zariski conjecture was considered for logarithmic vector fields on pairs and for logrithmic $1$-forms on pairs.
Posted Content
Algebraic integrability of foliations with numerically trivial canonical bundle.
Andreas Höring,Thomas Peternell +1 more
TL;DR: In this paper, the authors proved a flatness criterion under certain stability conditions for foliations with numerically trivial canonical bundles, which implies the algebraicity of leaves for sufficiently stable foliations such that the second Chern class does not vanish.
Journal ArticleDOI
Nonexistence for complete Kähler–Einstein metrics on some noncompact manifolds
TL;DR: In this article, it was shown that there is no complete Kahler-Einstein metric on a subvariety with codimension greater than or equal to 2, and a similar result for pairs.
References
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TL;DR: In this paper, the authors introduce the minimal model program and the canonical class of rational curves, and present the singularities of the model program, as well as three dimensional flops.
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Rational Curves on Algebraic Varieties
TL;DR: It is shown here how the model derived recently in [Bouchut-Boyaval, M3AS (23) 2013] can be modified for flows on rugous topographies varying around an inclined plane.