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Sep 21, 2013
09/13
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Sándor J Kovács
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The splitting principle states that morphisms in a derived category do not "split" accidentally. This has been successsfully applied in several characterizations of rational, DB, and other singularities. In this article I prove a general statement that implies many of the previous individual statements and improves some of the characterizations in the process.
Source: http://arxiv.org/abs/1108.1586v1
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Sep 18, 2013
09/13
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Sándor J. Kovács
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This is a survey on recent results regarding singularities that occur on higher dimensional stable varieties.
Source: http://arxiv.org/abs/1102.1240v2
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Sep 23, 2013
09/13
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Sándor J Kovács
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It is proved that for projective varieties having Du Bois singularities is equivalent to the condition that the coherent cohomology groups of the structure sheaf coincide with the appropriate Hodge components of the singular cohomology groups.
Source: http://arxiv.org/abs/1109.5569v2
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Sep 22, 2013
09/13
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Sándor J. Kovács
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The article has two parts. The first part is devoted to proving a singular version of the logarithmic Kodaira-Akizuki-Nakano vanishing theorem of Esnault and Viehweg. This is then used to prove other vanishing theorems. In the second part these vanishing theorems are used to prove an Arakelov-Parshin type boundedness result for families of canonically polarized varieties with rational Gorenstein singularities.
Source: http://arxiv.org/abs/math/0003019v4
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Sep 19, 2013
09/13
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Stefan Kebekus; Sandor J. Kovacs
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Let X be a projective variety which is covered by rational curves, for instance a Fano manifold over the complex numbers. In this setup, characterization and classification problems lead to the natural question: "Given two points on X, how many minimal degree rational curve are there which contain those points?". A recent answer to this question led to a number of new results in classi?cation theory. As an infinitesimal analogue, we ask "How many minimal degree rational curves...
Source: http://arxiv.org/abs/math/0206193v2
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Jun 27, 2018
06/18
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Sándor J Kovács; Zsolt Patakfalvi
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We prove a strengthening of Koll\'ar's Ampleness Lemma and use it to prove that any proper coarse moduli space of stable log-varieties of general type is projective. We also prove subadditivity of log-Kodaira dimension for fiber spaces whose general fiber is of log general type.
Topics: Algebraic Geometry, Mathematics
Source: http://arxiv.org/abs/1503.02952
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Jun 30, 2018
06/18
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Patrick Graf; Sándor J Kovács
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We investigate properties of potentially Du Bois singularities, that is, those that occur on the underlying space of a Du Bois pair. We show that a normal variety $X$ with potentially Du Bois singularities and Cartier canonical divisor $K_X$ is necessarily log canonical, and hence Du Bois. As an immediate corollary, we obtain the Lipman-Zariski conjecture for varieties with potentially Du Bois singularities. We also show that for a normal surface singularity, the notions of Du Bois and...
Topics: Mathematics, Algebraic Geometry
Source: http://arxiv.org/abs/1401.4976
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Sep 18, 2013
09/13
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Stefan Kebekus; Sandor J. Kovacs
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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...
Source: http://arxiv.org/abs/0707.2054v1
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Sep 19, 2013
09/13
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Sándor J Kovács; Karl Schwede
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This is a survey of some recent developments in the study of singularities related to the classification theory of algebraic varieties. In particular, the definition and basic properties of Du Bois singularities and their connections to the more commonly known singularities of the minimal model program are reviewed and discussed.
Source: http://arxiv.org/abs/0909.0993v1
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Sep 22, 2013
09/13
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Stefan Kebekus; Sandor J. Kovacs
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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 threefold Y that admits a non-constant map 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...
Source: http://arxiv.org/abs/0812.2305v1
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Sep 23, 2013
09/13
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Stefan Kebekus; Sandor J. Kovacs
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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 generally it has been conjectured by Viehweg that the base of a smooth family of canonically polarized varieties is of log general type if the family is of maximal variation. In this paper, we relate the variation of a family to the logarithmic Kodaira dimension of the base and give an affirmative...
Source: http://arxiv.org/abs/math/0511378v4
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Jul 20, 2013
07/13
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Sándor J Kovács; Karl Schwede
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Let $X$ be a variety and $H$ a Cartier divisor on $X$. We prove that if $H$ has Du Bois (or DB) singularities, then $X$ has Du Bois singularities near $H$. As a consequence, if $X \to S$ is a family over a smooth curve $S$ whose special fiber has Du Bois singularities, then the nearby fibers also have Du Bois singularities. We prove this by obtaining an injectivity theorem for certain maps of canonical modules. As a consequence, we also obtain a restriction theorem for certain non-lc ideals.
Source: http://arxiv.org/abs/1107.2349v3
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Sep 21, 2013
09/13
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János Kollár; Sándor J Kovács
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A recurring difficulty in the Minimal Model Program is that while log terminal singularities are quite well behaved (for instance, they are rational), log canonical singularities are much more complicated; they need not even be Cohen-Macaulay. The aim of this paper is to prove that log canonical singularities are Du Bois. The concept of Du Bois singularities, introduced by Steenbrink, is a weakening of rationality. We also prove flatness of the cohomology sheaves of the relative dualizing...
Source: http://arxiv.org/abs/0902.0648v4
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Jun 29, 2018
06/18
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Christopher D. Hacon; Sándor J Kovács
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The purpose of this note is to give a new proof of Alexeev's boundedness result for stable surfaces which is independent of the base field and to highlight some important consequences of this result.
Topics: Algebraic Geometry, Mathematics
Source: http://arxiv.org/abs/1603.04894
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Sep 23, 2013
09/13
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Christopher D Hacon; Sándor J Kovács
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We show that generic vanishing fails for singular varieties and in characteristic p>0.
Source: http://arxiv.org/abs/1212.5105v2
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Sep 23, 2013
09/13
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Christopher D. Hacon; Sándor J. Kovács
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It has been conjectured that varieties of general type do not admit nowhere vanishing holomorphic one-forms. We confirm this conjecture for smooth minimal varieties and for varieties whose Albanese variety is simple.
Source: http://arxiv.org/abs/math/0411049v1
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Sep 18, 2013
09/13
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Carolina Araujo; Stéphane Druel; Sándor J. Kovács
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We confirm Beauville's conjecture that claims that if the p-th exterior power of the tangent bundle of a smooth projective variety contains the p-th power of an ample line bundle, then the variety is either the projective space or the p-dimensional quadric hypersurface.
Source: http://arxiv.org/abs/0707.4310v2
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Sep 21, 2013
09/13
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Daniel Greb; Stefan Kebekus; Sándor J. Kovács
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Given a p-form defined on the smooth locus of a normal variety, and a resolution of singularities, we study the problem of extending the pull-back of the p-form over the exceptional set of the desingularization. For log canonical pairs and for certain values of p, we show that an extension always exists, possibly with logarithmic poles along the exceptional set. As a corollary, it is shown that sheaves of reflexive differentials enjoy good pull-back properties. A natural generalization of the...
Source: http://arxiv.org/abs/0808.3647v3
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Sep 17, 2013
09/13
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Daniel Greb; Stefan Kebekus; Sandor J. Kovacs; Thomas Peternell
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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...
Source: http://arxiv.org/abs/1003.2913v4
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Sep 23, 2013
09/13
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Sándor J. Kovács; Karl E. Schwede; Karen E. Smith
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We prove that a Cohen-Macaulay normal variety $X$ has Du Bois singularities if and only if $\pi_*\omega_{X'}(G) \simeq \omega_X$ for a log resolution $\pi: X' \to X$, where $G$ is the reduced exceptional divisor of $\pi$. Many basic theorems about Du Bois singularities become transparent using this characterization (including the fact that Cohen-Macaulay log canonical singularities are Du Bois). We also give a straightforward and self-contained proof that (generalizations of) semi-log-canonical...
Source: http://arxiv.org/abs/0801.1541v3
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Jun 30, 2018
06/18
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Thomas Bauer; Sándor J Kovács; Alex Küronya; Ernesto Carlo Mistretta; Tomasz Szemberg; Stefano Urbinati
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The aim of this note is to shed some light on the relationships among some notions of positivity for vector bundles that arose in recent decades. Our purpose is to study several of the positivity notions studied for vector bundles with some notions of asymptotic base loci that can be defined on the variety itself, rather than on the projectivization of the given vector bundle. We relate some of the different notions conjectured to be equivalent with the help of these base loci, and we show that...
Topics: Mathematics, Algebraic Geometry
Source: http://arxiv.org/abs/1406.5941
