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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Holomorphic symmetric differentials and a birational characterization of Abelian Varieties
TL;DR: In this paper, it was shown that a smooth projective variety is birational to an abelian variety if and only if it has Kodaira dimension 0 and some symmetric power of its cotangent sheaf is generically generated by its global sections.
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Rational endomorphisms of codimension one holomorphic foliations
TL;DR: In this article, dominant rational maps preserving singular holomorphic codimension one foliations on projective manifolds were studied and they exhibit non-trivial transverse dynamics.
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Inversion of adjunction for rational and Du Bois pairs
Sándor J. Kovács,Karl Schwede +1 more
TL;DR: In this article, the behavior of Du Bois singularities and rational pairs in families was studied in the non-pair setting, and it was shown that if a family over a smooth base has a special fiber with rational singularities, and if the general fiber has rational ones, then the total space has rational singularity near the boundary of the family.
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Uniformity for integral points on surfaces, positivity of log cotangent sheaves and hyperbolicity
TL;DR: In this article, it was shown that all subvarieties of a quasi-projective variety with positive log cotangent bundle are of log general type, and that smooth quasi projective varieties with positive and globally generated log cotsangent have finitely many integral points, generalizing a theorem of Moriwaki.
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On the MMP for rank one foliations on threefolds
Paolo Cascini,Calum Spicer +1 more
TL;DR: In this article, the existence of flips and the base point free theorem for log canonical foliated pairs of rank one on a Q-factorial projective klt was proved.
References
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Basic Algebraic Geometry
TL;DR: The second volume of Shafarevich's introductory book on algebraic geometry focuses on schemes, complex algebraic varieties and complex manifolds as discussed by the authors, and is suitable for beginning graduate students.
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The geometry of moduli spaces of sheaves
Daniel Huybrechts,Manfred Lehn +1 more
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Birational Geometry of Algebraic Varieties
János Kollár,Shigefumi Mori +1 more
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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Existence of minimal models for varieties of log general type
TL;DR: In this paper, it was shown that pl-flips exist in dimension n − 1, assuming finite generation in dimension N − 1 and assuming that pl flips exist in all dimensions.
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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.