Differential Forms on Log Canonical Spaces
TL;DR: 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.
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TL;DR: In this article, the flatness of leaves for sufficiently stable foliations with numerically trivial canonical bundles was proved under certain stability conditions, which implies the algebraicity of leaves in the case of minimal models with trivial canonical class.
Abstract: Given a reflexive sheaf on a mildly singular projective variety, we prove a flatness criterion under certain stability conditions. This implies the algebraicity of leaves for sufficiently stable foliations with numerically trivial canonical bundle such that the second Chern class does not vanish. Combined with the recent works of Druel and Greb–Guenancia–Kebekus this establishes the Beauville–Bogomolov decomposition for minimal models with trivial canonical class.
89 citations
TL;DR: In this article, it was shown that the Yau-Tian-Donaldson conjecture holds in the case of (possibly singular) canonically polarized (or quasi-projective) varieties.
Abstract: Let X be a canonically polarized variety, i.e. a complex projective variety such that its canonical class K (X) defines an ample -line bundle, and satisfying the conditions G (1) and S (2). Our main result says that X admits a Kahler-Einstein metric iff X has semi-log canonical singularities i.e. iff X is a stable variety in the sense of Kollar-Shepherd-Barron and Alexeev (whose moduli spaces are known to be compact). By definition a Kahler-Einstein metric in this singular context simply means a Kahler-Einstein on the regular locus of X with volume equal to the algebraic volume of K (X) , i.e. the top intersection number of K (X) . We also show that such a metric is uniquely determined and extends to define a canonical positive current in c (1)(K (X) ). Combined with recent results of Odaka our main result shows that X admits a Kahler-Einstein metric iff X is K-stable, which thus confirms the Yau-Tian-Donaldson conjecture in this general setting of (possibly singular) canonically polarized varieties. More generally, our results are shown to hold in the setting of log minimal varieties and they also generalize some prior results concerning Kahler-Einstein metrics on quasi-projective varieties.
86 citations
TL;DR: In this article, it was shown that for a quasi-projective variety X with only Kawamata log terminal singularities, there exists a Galois cover (GCC) ramified only over the singularities of X, such that the etale fundamental groups of X and Y agree.
Abstract: Given a quasi-projective variety X with only Kawamata log terminal singularities, we study the obstructions to extending finite etale covers from the smooth locus $X_{\mathrm{reg}}$ of $X$ to $X$ itself. A simplified version of our main results states that there exists a Galois cover $Y \rightarrow X$, ramified only over the singularities of $X$, such that the etale fundamental groups of $Y$ and of $Y_{\mathrm{reg}}$ agree. In particular, every etale cover of $Y_{\mathrm{reg}}$ extends to an etale cover of $Y$.
As first major application, we show that every flat holomorphic bundle defined on $Y_{\mathrm{reg}}$ extends to a flat bundle that is defined on all of $Y$. As a consequence, we generalise a classical result of Yau to the singular case: every variety with at worst terminal singularities and with vanishing first and second Chern class is a finite quotient of an Abelian variety. As a further application, we verify a conjecture of Nakayama and Zhang describing the structure of varieties that admit polarised endomorphisms.
83 citations
TL;DR: In this paper, it was shown that for a quasiprojective variety X with only Kawamata log terminal singularities, there exists a Galois cover Y→X, ramified only over the singularities of X, such that the etale fundamental groups of Y and of Yreg agree.
Abstract: Given a quasiprojective variety X with only Kawamata log terminal singularities, we study the obstructions to extending finite etale covers from the smooth locus Xreg of X to X itself. A simplified version of our main results states that there exists a Galois cover Y→X, ramified only over the singularities of X, such that the etale fundamental groups of Y and of Yreg agree. In particular, every etale cover of Yreg extends to an etale cover of Y. As a first major application, we show that every flat holomorphic bundle defined on Yreg extends to a flat bundle that is defined on all of Y. As a consequence, we generalize a classical result of Yau to the singular case: every variety with at worst terminal singularities and with vanishing first and second Chern class is a finite quotient of an abelian variety. As a further application, we verify a conjecture of Nakayama and Zhang describing the structure of varieties that admit polarized endomorphisms.
83 citations
TL;DR: In this article, the authors extend the Beauville-Bogomolov decomposition theorem to the singular setting and show that any complex projective variety of dimension at most five with canonical singularities and numerically trivial canonical class admits a finite cover, etale in codimension one, that decomposes as a product of an Abelian variety.
Abstract: In this paper we partly extend the Beauville–Bogomolov decomposition theorem to the singular setting. We show that any complex projective variety of dimension at most five with canonical singularities and numerically trivial canonical class admits a finite cover, etale in codimension one, that decomposes as a product of an Abelian variety, and singular analogues of irreducible Calabi–Yau and irreducible holomorphic symplectic varieties.
62 citations
References
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Book•
01 Jan 1974
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.
Abstract: The second volume of Shafarevich's introductory book on algebraic geometry focuses on schemes, complex algebraic varieties and complex manifolds. As with first volume the author has revised the text and added new material. Although the material is more advanced than in Volume 1 the algebraic apparatus is kept to a minimum making the book accessible to non-specialists. It can be read independently of the first volume and is suitable for beginning graduate students.
2,024 citations
Book•
01 Jan 1997TL;DR: In this paper, the Grauert-Mullich Theorem is used to define a moduli space for sheaves on K-3 surfaces, and the restriction of sheaves to curves is discussed.
Abstract: 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. Sheaves on Surfaces: 5. Construction methods 6. Moduli spaces on K3 surfaces 7. Restriction of sheaves to curves 8. Line bundles on the moduli space 9. Irreducibility and smoothness 10. Symplectic structures 11. Birational properties Glossary of notations References Index.
1,856 citations
Book•
01 Jan 1998TL;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.
Abstract: 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 7. Semi-stable minimal models.
1,754 citations
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.
Abstract: Assuming finite generation in dimension n − 1, we prove that pl-flips exist in dimension n.
1,612 citations
Book•
01 Jan 1996
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.
Abstract: I. Hilbert Schemes and Chow Varieties.- II. Curves on Varieties.- III. The Cone Theorem and Minimal Models.- IV. Rationally Connected Varieties.- V. Fano Varieties.- VI. Appendix.- References.
1,560 citations