1 - Oberwolfach Preprints (OWP)
http://publications.mfo.de/handle/mfo/19
The Oberwolfach Preprints (OWP) mainly contain research results related to a longer stay in Oberwolfach. In particular, this concerns the Research in Pairs program and the Oberwolfach Leibniz Fellows, but this can also include an Oberwolfach Lecture, for example.Mon, 10 Dec 2018 03:44:45 GMT2018-12-10T03:44:45ZCriteria for Algebraicity of Analytic Functions
http://publications.mfo.de/handle/mfo/1392
Criteria for Algebraicity of Analytic Functions
Bochnak, Jacek; Gwoździewicz, Janusz; Kucharz, Wojciech
We consider functions defined on an open subset of a nonsingular, either real or complex, algebraic set. We give criteria for an analytic function to be a Nash (resp. regular, resp. polynomial) function. Our criteria depend only on the behavior of such a function along irreducible nonsingular algebraic curves passing trough a given point. In the proofs we use results on algebraicity of formal power series, which are also established in this paper.
Mon, 12 Nov 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13922018-11-12T00:00:00ZBochnak, JacekGwoździewicz, JanuszKucharz, WojciechWe consider functions defined on an open subset of a nonsingular, either real or complex, algebraic set. We give criteria for an analytic function to be a Nash (resp. regular, resp. polynomial) function. Our criteria depend only on the behavior of such a function along irreducible nonsingular algebraic curves passing trough a given point. In the proofs we use results on algebraicity of formal power series, which are also established in this paper.Global Variants of Hartogs' Theorem
http://publications.mfo.de/handle/mfo/1391
Global Variants of Hartogs' Theorem
Bochnak, Jacek; Kucharz, Wojciech
Hartogs' theorem asserts that a separately holomorphic function, defined on an open subset of $\mathbb{C}^n$, is holomorphic in all the variables. We prove a global variant of this theorem for functions defined on an open subset of the product of complex algebraic manifolds. We also obtain global Hartogs-type theorems for complex Nash functions and complex regular functions.
Tue, 06 Nov 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13912018-11-06T00:00:00ZBochnak, JacekKucharz, WojciechHartogs' theorem asserts that a separately holomorphic function, defined on an open subset of $\mathbb{C}^n$, is holomorphic in all the variables. We prove a global variant of this theorem for functions defined on an open subset of the product of complex algebraic manifolds. We also obtain global Hartogs-type theorems for complex Nash functions and complex regular functions.Real Analyticity is Concentrated in Dimension 2
http://publications.mfo.de/handle/mfo/1390
Real Analyticity is Concentrated in Dimension 2
Bochnak, Jacek; Kucharz, Wojciech
We prove that a real-valued function on a real analytic manifold is analytic whenever all its restrictions to $2$-dimensional analytic submanifolds are analytic functions. We also obtain analogous results in the framework of Nash manifolds and nonsingular real algebraic sets. These results can be regarded as substitutes in the real case for the classical theorem of Hartogs, asserting that a complex-valued function defined on an open subset of $C^n$ is holomorphic if it is holomorphic with respect to each variable separately. In the proofs we use methods of real algebraic geometry even though the initial problem is purely analytic.
Mon, 05 Nov 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13902018-11-05T00:00:00ZBochnak, JacekKucharz, WojciechWe prove that a real-valued function on a real analytic manifold is analytic whenever all its restrictions to $2$-dimensional analytic submanifolds are analytic functions. We also obtain analogous results in the framework of Nash manifolds and nonsingular real algebraic sets. These results can be regarded as substitutes in the real case for the classical theorem of Hartogs, asserting that a complex-valued function defined on an open subset of $C^n$ is holomorphic if it is holomorphic with respect to each variable separately. In the proofs we use methods of real algebraic geometry even though the initial problem is purely analytic.Computing Congruence Quotients of Zariski Dense Subgroups
http://publications.mfo.de/handle/mfo/1389
Computing Congruence Quotients of Zariski Dense Subgroups
Detinko, Alla; Flannery, Dane; Hulpke, Alexander
We obtain a computational realization of the strong approximation theorem. That is, we develop algorithms to compute all congruence quotients modulo rational primes of a finitely generated Zariski dense group $H \leq \mathrm{SL}(n, \mathbb{Z})$ for $n \geq 2$. More generally, we are able to compute all congruence quotients of a finitely generated Zariski dense subgroup of $\mathrm{SL}(n, \mathbb{Q})$ for $n > 2$.
Fri, 26 Oct 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13892018-10-26T00:00:00ZDetinko, AllaFlannery, DaneHulpke, AlexanderWe obtain a computational realization of the strong approximation theorem. That is, we develop algorithms to compute all congruence quotients modulo rational primes of a finitely generated Zariski dense group $H \leq \mathrm{SL}(n, \mathbb{Z})$ for $n \geq 2$. More generally, we are able to compute all congruence quotients of a finitely generated Zariski dense subgroup of $\mathrm{SL}(n, \mathbb{Q})$ for $n > 2$.On the Invariants of the Cohomology of Complements of Coxeter Arrangements
http://publications.mfo.de/handle/mfo/1388
On the Invariants of the Cohomology of Complements of Coxeter Arrangements
Douglass, J. Matthew; Pfeiffer, Götz; Röhrle, Gerhard
We refine Brieskorn's study of the cohomology of the complement of the reflection arrangement of a finite Coxeter group W. As a result we complete the verification of a conjecture by Felder and Veselov that gives an explicit basis of the space of W-invariants
in this cohomology ring.
Mon, 22 Oct 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13882018-10-22T00:00:00ZDouglass, J. MatthewPfeiffer, GötzRöhrle, GerhardWe refine Brieskorn's study of the cohomology of the complement of the reflection arrangement of a finite Coxeter group W. As a result we complete the verification of a conjecture by Felder and Veselov that gives an explicit basis of the space of W-invariants
in this cohomology ring.Demailly’s Notion of Algebraic Hyperbolicity: Geometricity, Boundedness, Moduli of Maps
http://publications.mfo.de/handle/mfo/1387
Demailly’s Notion of Algebraic Hyperbolicity: Geometricity, Boundedness, Moduli of Maps
Javanpeykar, Ariyan; Kamenova, Ljudmila
Demailly's conjecture, which is a consequence of the Green-Griffiths-Lang conjecture on varieties of general type, states that an algebraically hyperbolic complex projective variety is Kobayashi hyperbolic. Our aim is to provide evidence for Demailly's conjecture by verifying several predictions it makes. We first define what an algebraically hyperbolic projective variety is, extending Demailly's definition to (not necessarily smooth) projective varieties over an arbitrary algebraically closed field of characteristic zero, and we prove that this property is stable under extensions of algebraically closed fields. Furthermore, we show that the set of (not necessarily surjective) morphisms from a projective variety Y to a projective algebraically hyperbolic variety X that map a fixed closed subvariety of Y onto a fixed closed subvariety of X is finite. As an application, we obtain that Aut(X) is finite and that every surjective endomorphism of X is an automorphism. Finally, we explore "weaker" notions of hyperbolicity related to boundedness of moduli spaces of maps, and verify similar predictions made by the Green-Griffiths-Lang conjecture on hyperbolic projective varieties.
Mon, 08 Oct 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13872018-10-08T00:00:00ZJavanpeykar, AriyanKamenova, LjudmilaDemailly's conjecture, which is a consequence of the Green-Griffiths-Lang conjecture on varieties of general type, states that an algebraically hyperbolic complex projective variety is Kobayashi hyperbolic. Our aim is to provide evidence for Demailly's conjecture by verifying several predictions it makes. We first define what an algebraically hyperbolic projective variety is, extending Demailly's definition to (not necessarily smooth) projective varieties over an arbitrary algebraically closed field of characteristic zero, and we prove that this property is stable under extensions of algebraically closed fields. Furthermore, we show that the set of (not necessarily surjective) morphisms from a projective variety Y to a projective algebraically hyperbolic variety X that map a fixed closed subvariety of Y onto a fixed closed subvariety of X is finite. As an application, we obtain that Aut(X) is finite and that every surjective endomorphism of X is an automorphism. Finally, we explore "weaker" notions of hyperbolicity related to boundedness of moduli spaces of maps, and verify similar predictions made by the Green-Griffiths-Lang conjecture on hyperbolic projective varieties.Affine Space Fibrations
http://publications.mfo.de/handle/mfo/1386
Affine Space Fibrations
Gurjar, Rajendra V.; Masuda, Kayo; Miyanishi, Masayoshi
We discuss various aspects of affine space fibrations. Our interest will be focused in the singular fibers, the generic fiber and the propagation of properties of a given smooth special fiber to nearby fibers.
Wed, 05 Sep 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13862018-09-05T00:00:00ZGurjar, Rajendra V.Masuda, KayoMiyanishi, MasayoshiWe discuss various aspects of affine space fibrations. Our interest will be focused in the singular fibers, the generic fiber and the propagation of properties of a given smooth special fiber to nearby fibers.Some Results Related to Schiffer's Problem
http://publications.mfo.de/handle/mfo/1385
Some Results Related to Schiffer's Problem
Kawohl, Bernd; Lucia, Marcello
We consider the following semilinear overdetermined problem on a two dimensional bounded or unbounded domain $\Omega$ with analytic boundary $\partial\Omega$ having at least one bounded connected component \begin{eqnarray*} \left\{ \begin{array}{l} - \Delta u = g(u) \quad \hbox{in } \Omega,\\ \frac{\partial u}{\partial \nu} =0 \, \hbox{ and } \, u = c \hbox{ on } \partial \Omega, \end{array} \right. \end{eqnarray*} where $c$ is a constant. When $g(c) =0$ the constant solution $u \equiv c$ is the unique solution. For $g(c) \not =0$, we show that the boundary is a circle if and only if the problem admits a solution that has constant third or fourth normal derivative along the boundary. A similar result involving the fifth normal derivative is proved.
Thu, 16 Aug 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13852018-08-16T00:00:00ZKawohl, BerndLucia, MarcelloWe consider the following semilinear overdetermined problem on a two dimensional bounded or unbounded domain $\Omega$ with analytic boundary $\partial\Omega$ having at least one bounded connected component \begin{eqnarray*} \left\{ \begin{array}{l} - \Delta u = g(u) \quad \hbox{in } \Omega,\\ \frac{\partial u}{\partial \nu} =0 \, \hbox{ and } \, u = c \hbox{ on } \partial \Omega, \end{array} \right. \end{eqnarray*} where $c$ is a constant. When $g(c) =0$ the constant solution $u \equiv c$ is the unique solution. For $g(c) \not =0$, we show that the boundary is a circle if and only if the problem admits a solution that has constant third or fourth normal derivative along the boundary. A similar result involving the fifth normal derivative is proved.Generalized Vector Cross Products and Killing Forms on Negatively Curved Manifolds
http://publications.mfo.de/handle/mfo/1377
Generalized Vector Cross Products and Killing Forms on Negatively Curved Manifolds
Barberis, María Laura; Moroianu, Andrei; Semmelmann, Uwe
Motivated by the study of Killing forms on compact Riemannian manifolds of negative sectional curvature, we introduce the notion of generalized vector cross products on $\mathbb{R}^n$ and give their classification. Using previous results about Killing tensors on negatively curved manifolds and a new characterization of $\mathrm{SU}(3)$-structures in dimension $6$ whose associated $3$-form is Killing, we then show that every Killing $3$-form on a compact $n$-dimensional Riemannian manifold with negative sectional curvature vanishes if $n\ge 4$.
Tue, 17 Jul 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13772018-07-17T00:00:00ZBarberis, María LauraMoroianu, AndreiSemmelmann, UweMotivated by the study of Killing forms on compact Riemannian manifolds of negative sectional curvature, we introduce the notion of generalized vector cross products on $\mathbb{R}^n$ and give their classification. Using previous results about Killing tensors on negatively curved manifolds and a new characterization of $\mathrm{SU}(3)$-structures in dimension $6$ whose associated $3$-form is Killing, we then show that every Killing $3$-form on a compact $n$-dimensional Riemannian manifold with negative sectional curvature vanishes if $n\ge 4$.Metric Connections with Parallel Skew-Symmetric Torsion
http://publications.mfo.de/handle/mfo/1376
Metric Connections with Parallel Skew-Symmetric Torsion
Cleyton, Richard; Moroianu, Andrei; Semmelmann, Uwe
A geometry with parallel skew-symmetric torsion is a Riemannian manifold carrying a metric connection with parallel skew-symmetric torsion. Besides the trivial case of the Levi-Civita connection, geometries with non-vanishing parallel skew-symmetric torsion arise naturally in several geometric contexts, e.g. on naturally reductive homogeneous spaces, nearly Kähler or nearly parallel G2-manifolds, Sasakian and 3-Sasakian manifolds, or twistor spaces over quaternion-Kähler manifolds with positive scalar curvature. In this paper we study the local structure of Riemannian manifolds carrying a metric connection with parallel skew-symmetric torsion. On every such manifold one can define a natural splitting of the tangent bundle which gives rise to a Riemannian submersion over a geometry with parallel skew-symmetric torsion of smaller dimension endowed with some extra structure. We show how previously known examples of geometries with parallel skew-symmetric torsion fit into this pattern, and construct several new examples. In the particular case where the above Riemannian submersion has the structure of a principal bundle, we give the complete local classification of the corresponding geometries with parallel skew-symmetric torsion.
Mon, 16 Jul 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13762018-07-16T00:00:00ZCleyton, RichardMoroianu, AndreiSemmelmann, UweA geometry with parallel skew-symmetric torsion is a Riemannian manifold carrying a metric connection with parallel skew-symmetric torsion. Besides the trivial case of the Levi-Civita connection, geometries with non-vanishing parallel skew-symmetric torsion arise naturally in several geometric contexts, e.g. on naturally reductive homogeneous spaces, nearly Kähler or nearly parallel G2-manifolds, Sasakian and 3-Sasakian manifolds, or twistor spaces over quaternion-Kähler manifolds with positive scalar curvature. In this paper we study the local structure of Riemannian manifolds carrying a metric connection with parallel skew-symmetric torsion. On every such manifold one can define a natural splitting of the tangent bundle which gives rise to a Riemannian submersion over a geometry with parallel skew-symmetric torsion of smaller dimension endowed with some extra structure. We show how previously known examples of geometries with parallel skew-symmetric torsion fit into this pattern, and construct several new examples. In the particular case where the above Riemannian submersion has the structure of a principal bundle, we give the complete local classification of the corresponding geometries with parallel skew-symmetric torsion.