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Stefan Kebekus

Publications and source records attributed to Stefan Kebekus.

At least 37 records · Page 2Linked to original sources
arXiv2015

Differential forms in positive characteristic avoiding resolution of singularities

This paper studies several notions of sheaves of differential forms that are better behaved on singular varieties than Kähler differentials. Our main focus lies on varieties that are defined over fields of positive characteristic. We identify two promising notions: the sheafification with respect to the cdh-topology, and right Kan extension from the subcategory of smooth varieties to the category of all varieties. Our main results are that both are cdh-sheaves and agree with Kähler differentials on smooth varieties. They agree on all varieties under weak resolution of singularities. A number of examples highlight the difficulties that arise with torsion forms and with alternative candiates.

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arXiv2014

Comparing A^1-h-cobordism and A^1-weak equivalence

We study the problem of classifying projectivizations of rank-two vector bundles over ${\mathbb P}^2$ up to various notions of equivalence that arise naturally in ${\mathbb A}^1$-homotopy theory, namely ${\mathbb A}^1$-weak equivalence and ${\mathbb A}^1$-$h$-cobordism. First, we classify such varieties up to ${\mathbb A}^1$-weak equivalence: over algebraically closed fields having characteristic unequal to two the classification can be given in terms of characteristic classes of the underlying vector bundle. When the base field is ${\mathbb C}$, this classification result can be compared to a corresponding topological result and we find that the algebraic and topological homotopy classifications agree. Second, we study the problem of classifying such varieties up to ${\mathbb A}^1$-$h$-cobordism using techniques of deformation theory. To this end, we establish a deformation rigidity result for ${\mathbb P}^1$-bundles over ${\mathbb P}^2$ which links ${\mathbb A}^1$-$h$-cobordisms to deformations of the underlying vector bundles. Using results from the deformation theory of vector bundles we show that if $X$ is a ${\mathbb P}^1$-bundle over ${\mathbb P}^2$ and $Y$ is the projectivization of a direct sum of line bundles on ${\mathbb P}^2$, then if $X$ is ${\mathbb A}^1$-weakly equivalent to $Y$, $X$ is also ${\mathbb A}^1$-$h$-cobordant to $Y$. Finally, we discuss some subtleties inherent in the definition of ${\mathbb A}^1$-$h$-cobordism. We show, for instance, that direct ${\mathbb A}^1$-$h$-cobordism fails to be an equivalence relation.

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arXiv2013

Etale covers of Kawamata log terminal spaces and their smooth loci

Working with a singular variety X, one is often interested in comparing the set of etale covers of X with that of its smooth locus X_reg. More precisely, one may ask: What are the obstructions to extend finite etale covers of X_reg to all of X? How do the etale fundamental groups of X and of its smooth locus compare? For projective varieties with Kawamata log terminal (klt) singularities we answer these questions in part. The main result of this paper asserts that there are no infinite towers of finite morphisms over a klt base variety where all morphisms are etale in codimension one, but branched over a small set. In a certain sense, this result can be seen as saying that the difference between the etale covers of X and those of X_reg is finite if X is klt. As an immediate application, we construct a finite covering Y of X, etale in codimension one, such that the etale fundamental groups of Y and Y_reg agree.

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arXiv2013

Pull-back Morphisms for Reflexive Differential Forms

Let f : X -> Y be a morphism between normal complex varieties, and assume that Y is Kawamata log terminal. Given any differential form, defined on the smooth locus of Y, we construct a "pull-back form" on X. The pull-back map obtained by this construction is O_Y-linear, uniquely determined by natural universal properties and exists even in cases where the image of f is entirely contained in the singular locus of the target variety Y. One relevant setting covered by the construction is that where f is the inclusion (or normalisation) of the singular locus of Y. As an immediate corollary, we show that differential forms defined on the smooth locus of Y induce forms on every stratum of the singularity stratification. The same result also holds for many Whitney stratifications.

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arXiv2012

Reflexive differential forms on singular spaces -- Geometry and Cohomology

Based on a recent extension theorem for reflexive differential forms, that is, regular differential forms defined on the smooth locus of a possibly singular variety, we study the geometry and cohomology of sheaves of reflexive differentials. First, we generalise the extension theorem to holomorphic forms on locally algebraic complex spaces. We investigate the (non-)existence of reflexive pluri-differentials on singular rationally connected varieties, using a semistability analysis with respect to movable curve classes. The necessary foundational material concerning this stability notion is developed in an appendix to the paper. Moreover, we prove that Kodaira-Akizuki-Nakano vanishing for sheaves of reflexive differentials holds in certain extreme cases, and that it fails in general. Finally, topological and Hodge-theoretic properties of reflexive differentials are explored.

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arXiv2012

Singular spaces with trivial canonical class

The classical Beauville-Bogomolov Decomposition Theorem asserts that any compact Kähler manifold with numerically trivial canonical bundle admits an étale cover that decomposes into a product of a torus, and irreducible, simply-connected Calabi-Yau-- and holomorphic-symplectic manifolds. The decomposition of the simply-connected part corresponds to a decomposition of the tangent bundle into a direct sum whose summands are integrable and stable with respect to any polarisation. Building on recent extension theorems for differential forms on singular spaces, we prove an analogous decomposition theorem for the tangent sheaf of projective varieties with canonical singularities and numerically trivial canonical class. In view of recent progress in minimal model theory, this result can be seen as a first step towards a structure theory of manifolds with Kodaira dimension zero. Based on our main result, we argue that the natural building blocks for any structure theory are two classes of canonical varieties, which generalise the notions of irreducible Calabi-Yau-- and irreducible holomorphic-symplectic manifolds, respectively.

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arXiv2011

Differential Forms on Log Canonical Spaces

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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arXiv2011

Deformation along subsheaves, II

Let Y be a compact reduced subspace of a complex manifold X, and let F be a subsheaf of the tangent bundle T_X which is closed under the Lie bracket. This paper discusses criteria to guarantee that infinitesimal deformations of the inclusion morphism Y -> X give rise to positive-dimensional deformation families, deforming the inclusion map "along the sheaf F". In case where X is complex-symplectic and F is the sheaf of Hamiltonian vector fields, this partially reproduces known results on unobstructedness of deformations of Lagrangian submanifolds. Written for the IMPANGA Lecture Notes series, this paper aims at simplicity and clarity of argument. It does not strive to present the shortest proofs or most general results available. The proof is rather elementary and geometric, constructing higher-order liftings of a given infinitesimal deformation using flow maps of carefully crafted time-dependent vector fields.

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arXiv2010

Families over special base manifolds and a conjecture of Campana

Consider a smooth, projective family of canonically polarized varieties over a smooth, quasi-projective base manifold Y, all defined over the complex numbers. It has been conjectured that the family is necessarily isotrivial if Y is special in the sense of Campana. We prove the conjecture when Y is a surface or threefold. The proof uses sheaves of symmetric differentials associated to fractional boundary divisors on log canonical spaces, as introduced by Campana in his theory of Orbifoldes Geometriques. We discuss a weak variant of the Harder-Narasimhan Filtration and prove a version of the Bogomolov-Sommese Vanishing Theorem that take the additional fractional positivity along the boundary into account. A brief, but self-contained introduction to Campana's theory is included for the reader's convenience.

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arXiv2010

Deformations along subsheaves

Let f : Y -> X be a morphism of complex projective manifolds, and let F be a subsheaf of the tangent bundle which is closed under the Lie bracket, but not necessarily a foliation. This short paper contains an elementary and very geometric argument to show that all obstructions to deforming the morphism f along the sheaf F lie in the first cohomology group H^1(Y, F_Y) of the sheaf F_Y, which is the image of f^*(F) in f^*(T_X) under the pull-back of the inclusion map. Special cases of this result include the theory of deformation along a (possibly singular) foliation, logarithmic deformation theory and deformations with fixed points.

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arXiv2009

Extension theorems for differential forms and Bogomolov-Sommese vanishing on log canonical varieties

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 well-known Bogomolov-Sommese vanishing theorem to log canonical threefold pairs follows.

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arXiv2009

Positive sheaves of differentials coming from coarse moduli spaces

Consider a smooth projective family of canonically polarized complex manifolds over a smooth quasi-projective complex base U, and suppose the family is non-isotrivial. If Y is a smooth compactification of U, such that D := Y U is a simple normal crossing divisor, then we can consider the sheaf of differentials with logarithmic poles along D. Viehweg and Zuo have shown that for some number m>0, the m-th symmetric power of this sheaf admits many sections. More precisely, the m-th symmetric power contains an invertible sheaf whose Kodaira-Iitaka dimension is at least the variation of the family. We refine this result and show that this "Viehweg-Zuo sheaf" comes from the coarse moduli space associated to the given family, at least generically. As an immediate corollary, if U is a surface, we see that the non-isotriviality assumption implies that U cannot be special in the sense of Campana.

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arXiv2008

The structure of surfaces and threefolds mapping to the moduli stack of canonically polarized varieties

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 program of Y: in all relevant cases the minimal model program leads to a fiber space whose fibration factors the moduli map. A much refined affirmative answer to Viehweg's conjecture for families over threefolds follows as a corollary. For families over surfaces, the moduli map can be often be described quite explicitly. Slightly weaker results are obtained for families of varieties with trivial, or more generally semi-ample canonical bundle.

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arXiv2007

The structure of surfaces mapping to the moduli stack of canonically polarized varieties

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 result, we can describe the fibration induced by the moduli map quite explicitly. A refined affirmative answer to Viehweg's conjecture for families over surfaces follows as a corollary.

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arXiv2007

Endomorphisms of projective varieties

We study complex projective manifolds X that admit surjective endomorphisms f:X->X of degree at least two. In case f is etale, we prove structure theorems that describe X. In particular, a rather detailed description is given if X is a uniruled threefold. As to the ramified case, we first prove a general theorem stating that the vector bundle associated to a Galois covering of projective manifolds is ample (resp. nef) under very mild conditions. This is applied to the study of ramified endomorphisms of Fano manifolds with second Betti number one. It is conjectured that the projective space is the only Fano manifold admitting admitting an endomorphism of degree d>1, and we prove that in several cases. A part of the argumentation is based on a new characterization of the projective space as the only manifold that admits an ample subsheaf in its tangent bundle.

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arXiv2007

Families of varieties of general type over compact bases

Let f: X -> Y be a smooth family of canonically polarized complex varieties over a smooth base. Generalizing the classical Shafarevich hyperbolicity conjecture, Viehweg conjectured that Y is necessarily of log general type if the family has maximal variation. A somewhat stronger and more precise version of Viehweg's conjecture was shown by the authors in arXiv:math/0511378 in the case where Y is a quasi-projective surface. Assuming that the minimal model program holds, this very short paper proves the same result for projective base manifolds Y of arbitrary dimension.

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arXiv2006

Families of canonically polarized varieties over surfaces

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 answer to Viehweg's conjecture for families parametrized by surfaces.

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