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Showing papers in "Mathematische Zeitschrift in 2013"


Journal ArticleDOI
TL;DR: In this article it was shown that the Jordan decomposition of characters introduced by Lusztig and the generalized Harish-Chandra theory of unipotent characters due to Broue-Malle-Michel are equivariant with respect to group automorphisms under the additional assumption that the linear algebraic group associated to the character has connected center.
Abstract: We show that several character correspondences for finite reductive groups $$G$$ are equivariant with respect to group automorphisms under the additional assumption that the linear algebraic group associated to $$G$$ has connected center. The correspondences we consider are the so-called Jordan decomposition of characters introduced by Lusztig and the generalized Harish-Chandra theory of unipotent characters due to Broue–Malle–Michel. In addition we consider a correspondence giving character extensions, due to the second author, in order to verify the inductive McKay condition from Isaacs–Malle–Navarro for the non-abelian finite simple groups of Lie types $$^3\mathsf{D }_4,\mathsf{E }_8,\mathsf{F }_4,^2\mathsf{F }_4$$ , and $$\mathsf{G }_2$$ .

71 citations


Journal ArticleDOI
TL;DR: In this paper, a dynamical system defined on a subset of the variety of Lie algebras, parameterizing the space of all simply connected homogeneous spaces of dimension ρ ≥ 0, with a ρ-dimensional isotropy, which is proved to be equivalent in a precise sense to the Ricci flow, is presented.
Abstract: We present in this paper a general approach to study the Ricci flow on homogeneous manifolds. Our main tool is a dynamical system defined on a subset $$\mathcal H _{q,n}$$ of the variety of $$(q+n)$$ -dimensional Lie algebras, parameterizing the space of all simply connected homogeneous spaces of dimension $$n$$ with a $$q$$ -dimensional isotropy, which is proved to be equivalent in a precise sense to the Ricci flow. The approach is useful to better visualize the possible (nonflat) pointed limits of Ricci flow solutions, under diverse rescalings, as well as to determine the type of the possible singularities. Ancient solutions arise naturally from the qualitative analysis of the evolution equation. We develop two examples in detail: a $$2$$ -parameter subspace of $$\mathcal H _{1,3}$$ reaching most of $$3$$ -dimensional geometries, and a $$2$$ -parameter family in $$\mathcal H _{0,n}$$ of left-invariant metrics on $$n$$ -dimensional compact and non-compact semisimple Lie groups.

57 citations


Journal ArticleDOI
TL;DR: In this paper, the authors used the notion of multi-Reedy categories to prove that, if a C is a Reedy category, then C is also a reedy category.
Abstract: We use the notion of multi-Reedy category to prove that, if $$\mathcal C $$ is a Reedy category, then $$\varTheta \mathcal C $$ is also a Reedy category. This result gives a new proof that the categories $$\varTheta _n$$ are Reedy categories. We then define elegant Reedy categories, for which we prove that the Reedy and injective model structures coincide.

51 citations


Journal ArticleDOI
TL;DR: In this paper, a general uniruledness theorem for base loci of adjoint divisors of the Minimal Model Program (MMP) has been shown to hold.
Abstract: We explain how to deduce from recent results in the Minimal Model Program a general uniruledness theorem for base loci of adjoint divisors. As a special case, we recover previous results by Takayama.

49 citations


Journal ArticleDOI
TL;DR: In this article, the question of whether hypercyclicity is sufficient for distributional chaos for a continuous linear operator was answered in the negative and it was shown that mixing property does not suffice.
Abstract: In this article we answer in the negative the question of whether hypercyclicity is sufficient for distributional chaos for a continuous linear operator (we even prove that the mixing property does not suffice). Moreover, we show that an extremal situation is possible: There are (hypercyclic and non-hypercyclic) operators such that the whole space consists, except zero, of distributionally irregular vectors.

49 citations


Journal ArticleDOI
TL;DR: In this paper, the authors consider Ricci flow of complete Riemannian manifolds with bounded nonnegative curvature operator, non-zero asymptotic volume ratio and no boundary and prove scale invariant estimates for these solutions.
Abstract: We consider Ricci flow of complete Riemannian manifolds which have bounded non-negative curvature operator, non-zero asymptotic volume ratio and no boundary. We prove scale invariant estimates for these solutions. Using these estimates, we show that there is a limit solution, obtained by scaling down this solution at a fixed point in space. This limit solution is an expanding soliton coming out of the asymptotic cone at infinity.

48 citations


Journal ArticleDOI
TL;DR: In this article, the Cheeger-Muller/Bismut-Zhang theorem for manifold-swithboundary and the gluing-formulae-of-theanalytictorsionofflat vector bundles in full generality was derived.
Abstract: In this paper, we derive the Cheeger-Muller/Bismut-Zhang theorem for manifoldswithboundaryandthegluingformulafortheanalytictorsionofflatvectorbundles in full generality, i.e., we do not assume that the Hermitian metric on the flat vector bundle is flat nor that the Riemannian metric has product structure near the boundary.

48 citations


Journal ArticleDOI
TL;DR: In this paper, the connections between volume growth, spectral properties and stochastic completeness of locally finite weighted graphs were studied and a condition was given for a class of graphs with a very weak spherical symmetry.
Abstract: We study the connections between volume growth, spectral properties and stochastic completeness of locally finite weighted graphs. For a class of graphs with a very weak spherical symmetry we give a condition which implies both stochastic incompleteness and discreteness of the spectrum. We then use these graphs to give some comparison results for both stochastic completeness and estimates on the bottom of the spectrum for general locally finite weighted graphs.

47 citations


Journal ArticleDOI
TL;DR: In this paper, an explicit and geometrically meaningful formula for the heat kernel of the conformal sub-Laplacian of the CR sphere has been obtained, where the key point is to work in a set of coordinates that reflect the symmetries coming from the fibration.
Abstract: We study the heat kernel of the sub-Laplacian \(L\) on the CR sphere \(\mathbb{S }^{2n+1}\). An explicit and geometrically meaningful formula for the heat kernel is obtained. As a by-product we recover in a simple way the Green function of the conformal sub-Laplacian \(-L+n^2\) that was obtained by Geller (J Differ Geom 15:417–435, 1980), and also get an explicit formula for the sub-Riemannian distance. The key point is to work in a set of coordinates that reflects the symmetries coming from the fibration \(\mathbb{S }^{2n+1} \rightarrow \mathbb{CP }^n\).

46 citations


Journal ArticleDOI
TL;DR: In this article, a general explicit formula for the matrix coefficients of the Weil representation of the modular group is given for the case when the representation factors through a non-degenerate quadratic form.
Abstract: To any finite quadratic module, that is, a finite abelian group together with a non-degenerate quadratic form, it is possible to associate a representation of \(\mathrm{Mp}_{2}(\mathbb Z )\), the metaplectic cover of the modular group This representation is usually referred to as a Weil representation and our main result is a general explicit formula for its matrix coefficients This result completes earlier work by Scheithauer in the case when the representation factors through \(\mathrm{SL}_{2}(\mathbb Z )\) Furthermore, our formula is given in a such a way that it is easy to implement efficiently on a computer

45 citations


Journal ArticleDOI
TL;DR: In this article, the Dolbeault cohomology of direct sums of holomorphic line bundles over G/Γ is computed for semi-direct products of Lie groups with lattices Γ such that N are nilpotent Lie groups.
Abstract: We consider semi-direct products \({\mathbb{C}^{n}\ltimes_{\phi}N}\) of Lie groups with lattices Γ such that N are nilpotent Lie groups with left-invariant complex structures. We compute the Dolbeault cohomology of direct sums of holomorphic line bundles over G/Γ by using the Dolbeaut cohomology of the Lie algebras of the direct product \({\mathbb{C}^{n}\times N}\) . As a corollary of this computation, we can compute the Dolbeault cohomology Hp,q(G/Γ) of G/Γ by using a finite dimensional cochain complexes. Computing some examples, we observe that the Dolbeault cohomology varies for choices of lattices Γ.

Journal ArticleDOI
TL;DR: In this article, it was shown that for torus manifolds of dimension greater than six, there are infinitely many conjugacy classes and that the fundamental groups of locally standard torus manifold groups are not homeomorphic.
Abstract: In 2006 Masuda and Suh asked if two compact non-singular toric varieties having isomorphic cohomology rings are homeomorphic. In the first part of this paper we discuss this question for topological generalizations of toric varieties, so-called torus manifolds. For example we show that there are homotopy equivalent torus manifolds which are not homeomorphic. Moreover, we characterize those groups which appear as the fundamental groups of locally standard torus manifolds. In the second part we give a classification of quasitoric manifolds and certain six-dimensional torus manifolds up to equivariant diffeomorphism. In the third part we enumerate the number of conjugacy classes of tori in the diffeomorphism group of torus manifolds. For torus manifolds of dimension greater than six there are always infinitely many conjugacy classes. We give examples which show that this does not hold for six-dimensional torus manifolds.

Journal ArticleDOI
TL;DR: In this paper, a relation between the superrigidity of Fano manifold and its slope stability was proved, in the sense of Ross and Thomas (J Alg Geom 16:201-205, 2007).
Abstract: We prove a relation between birational superrigidity of Fano manifold and its slope stability in the sense of Ross and Thomas (J Alg Geom 16:201–205, 2007).

Journal ArticleDOI
TL;DR: In this article, Schrodinger operators on possibly noncompact Riemannian manifolds, acting on sections in vector bundles, with locally square integrable potentials whose negative part is in the underlying Kato class, are shown to be essentially self-adjoint under geodesic completeness.
Abstract: We consider Schrodinger operators on possibly noncompact Riemannian manifolds, acting on sections in vector bundles, with locally square integrable potentials whose negative part is in the underlying Kato class. Using path integral methods, we prove that under geodesic completeness these differential operators are essentially self-adjoint on \(\mathsf{C }^{\infty }_0\), and that the corresponding operator closures are semibounded from below. These results apply to nonrelativistic Pauli–Dirac operators that describe the energy of Hydrogen type atoms on Riemannian \(3\)-manifolds.

Journal ArticleDOI
TL;DR: In this paper, Zhang and Zhu give some applications of the Bochner type formula on Alexandrov spaces, and obtain (sharp) Li-Yau's estimate for positve solutions of heat equations.
Abstract: In the previous work (Zhang and Zhu in J Differ Geom, http://arxiv.org/pdf/1012.4233v3 , 2012), the second and third authors established a Bochner type formula on Alexandrov spaces. The purpose of this paper is to give some applications of the Bochner type formula. Firstly, we extend the sharp lower bound estimates of spectral gap, due to Chen and Wang (Sci Sin (A) 37:1–14, 1994), Chen and Wang (Sci Sin (A) 40:384–394, 1997) and Bakry–Qian (Adv Math 155:98–153, 2000), from smooth Riemannian manifolds to Alexandrov spaces. As an application, we get an Obata type theorem for Alexandrov spaces. Secondly, we obtain (sharp) Li–Yau’s estimate for positve solutions of heat equations on Alexandrov spaces.

Journal ArticleDOI
TL;DR: In this article, the notion of toric order was introduced and all toric orders which are 3-dimensional Calabi-Yau algebras can be constructed from dimer models on a torus.
Abstract: Dimer models have been used in string theory to construct path algebras with relations that are 3-dimensional Calabi–Yau Algebras. These constructions result in algebras that share some specific properties: they are finitely generated modules over their centers and their representation spaces are toric varieties. In order to describe these algebras we introduce the notion of a toric order and show that all toric orders which are 3-dimensional Calabi–Yau algebras can be constructed from dimer models on a torus. Toric orders are examples of a much broader class of algebras: positively graded cancellation algebras. For these algebras the CY-3 condition implies the existence of a weighted quiver polyhedron, which is an extension of dimer models obtained by replacing the torus with any two-dimensional compact orientable orbifold.

Journal ArticleDOI
TL;DR: In this article, the Hardy-Littlewood-Sobolev inequality was used to obtain an integrability result for positive solutions of a class of Choquard type equations, which is equivalent to integral systems involving the Bessel potential and the Riesz potential.
Abstract: This paper is concerned with positive solutions of a class of Choquard type equations. Such equations are equivalent to integral systems involving the Bessel potential and the Riesz potential. By using two regularity lifting lemmas introduced by Chen and Li [2], we study the regularity for integrable solutions u. We first use the Hardy–Littlewood–Sobolev inequality to obtain an integrability result. Then, it is improved to \({u \in L^s(R^n)}\) for all \({s \in [1, \infty]}\) by an iteration. Next, we use the properties of the contraction map and the shrinking map to prove that u is Lipschitz continuous. Finally, we establish the smoothness of u by a bootstrap argument. Our technique can also be used to handle other integral systems involving the Riesz potential or the Bessel potential, such as the Hartree type equations.

Journal ArticleDOI
TL;DR: In this article, the sign change problem of a cusp form of half integral weight whose Fourier coefficients are all real is studied, and lower bounds of the best possible order of magnitude are established for the number of coefficients that have the same signs.
Abstract: Let \({{\mathfrak f}}\) be a cusp form of half integral weight whose Fourier coefficients \({{\mathfrak a}_{\mathfrak f}(n)}\) are all real. We study the sign change problem of \({{\mathfrak a}_{\mathfrak f}(n)}\) , when n runs over some specific sets of integers. Lower bounds of the best possible order of magnitude are established for the number of those coefficients that have the same signs. These give an improvement on some recent results of Bruinier and Kohnen (Modular forms on Schiermonnikoong. Cambridge University Press, Cambridge 57–66, 2008) and Kohnen (Int. J. Number. Theory 6:1255–1259, 2010).

Journal ArticleDOI
TL;DR: In this article, it was shown that the abelian category of coherent functors over a contravariantly finite rigid subcategory in a triangulated category is equivalent to the Gabriel-Zisman localization at the class of regular maps of a certain factor category.
Abstract: We show that the abelian category \(\mathsf{mod}\text{-}\mathcal{X }\) of coherent functors over a contravariantly finite rigid subcategory \(\mathcal{X }\) in a triangulated category \(\mathcal{T }\) is equivalent to the Gabriel–Zisman localization at the class of regular maps of a certain factor category of \(\mathcal{T }\), and moreover it can be calculated by left and right fractions. Thus we generalize recent results of Buan and Marsh. We also extend recent results of Iyama–Yoshino concerning subfactor triangulated categories arising from mutation pairs in \(\mathcal{T }\). In fact we give a classification of thick triangulated subcategories of a natural pretriangulated factor category of \(\mathcal{T }\) and a classification of functorially finite rigid subcategories of \(\mathcal{T }\) if the latter has Serre duality. In addition we characterize \(2\)-cluster tilting subcategories along these lines. Finally we extend basic results of Keller–Reiten concerning the Gorenstein and the Calabi–Yau property for categories arising from certain rigid, not necessarily cluster tilting, subcategories, as well as several results of the literature concerning the connections between \(2\)-cluster tilting subcategories of triangulated categories and tilting subcategories of the associated abelian category of coherent functors.

Journal ArticleDOI
TL;DR: In this paper, a new characterization of freeness in which the second Betti number of the arrangement plays a crucial role is presented, in which it is shown that the multirestriction of a free arrangement is also free.
Abstract: Ziegler showed that the multirestriction of a free arrangement is also free. After Ziegler’s work, several results concerning the “reverse direction”, i.e., characterizing freeness of an arrangement via that of its multirestriction, have appeared. In this paper, we prove a new characterization of freeness in which the second Betti number of the arrangement plays a crucial role.

Journal ArticleDOI
TL;DR: In this article, the Trotter property of locally convex Lie groups has been studied in the context of automorphisms of principal bundles over compact smooth manifolds, and it has been shown that for smooth (resp., analytic) unitary representations of Frechet-Lie supergroups, the common domain of the k-fold products of the operators can be extended to the space of analytic vectors for G.
Abstract: A locally convex Lie group G has the Trotter property if, for every \(x_1, x_2 \in \mathfrak{g }\), $$\begin{aligned} \exp _G(t(x_1 + x_2))=\lim _{n \rightarrow \infty } \left(\exp _G\left(\frac{t}{n}x_1\right)\exp _G\left(\frac{t}{n}x_2\right)\right)^n \end{aligned}$$ holds uniformly on compact subsets of \(\mathbb{R }\). All locally exponential Lie groups have this property, but also groups of automorphisms of principal bundles over compact smooth manifolds. A key result of the present article is that, if G has the Trotter property, \(\pi : G \rightarrow {\mathrm{GL}}(V)\) is a continuous representation of G on a locally convex space, and \(v \in V\) is a vector such that \(\overline{\mathtt{d}\pi }(x)v :=\frac{d}{dt}|_{t=0} \pi (\exp _G(tx))v\) exists for every \(x \in \mathfrak{g }\), then the map \(\mathfrak{g }\rightarrow V,x \mapsto \overline{\mathtt{d}\pi }(x)v\) is linear. Using this result we conclude that, for a representation of a locally exponential Frechet–Lie group G on a metrizable locally convex space, the space of \(\mathcal{C }^{k}\)-vectors coincides with the common domain of the k-fold products of the operators \(\overline{\mathtt{d}\pi }(x)\). For unitary representations on Hilbert spaces, the assumption of local exponentiality can be weakened to the Trotter property. As an application, we show that for smooth (resp., analytic) unitary representations of Frechet–Lie supergroups \((G,\mathfrak{g })\) where G has the Trotter property, the common domain of the operators of \(\mathfrak{g }=\mathfrak{g }_{\overline{0}}\oplus \mathfrak{g }_{\overline{1}}\) can always be extended to the space of smooth (resp., analytic) vectors for G.

Journal ArticleDOI
TL;DR: In this paper, it was shown that the quotient C4/G admits a symplectic resolution for the dihedral group of order eight and the quaternionic group in order eight.
Abstract: We show that the quotient C4/G admits a symplectic resolution for \({G = Q_8 \times_{{\bf Z}/2} D_8 < {\sf Sp}_4({\bf C})}\). Here Q8 is the quaternionic group of order eight and D8 is the dihedral group of order eight, and G is the quotient of their direct product which identifies the nontrivial central elements −Id of each. It is equipped with the tensor product representation \({{\bf C}^2 \boxtimes {\bf C}^2 \cong {\bf C}^4}\). This group is also naturally a subgroup of the wreath product group \({Q_8^2 \rtimes S_2 < {\sf Sp}_4({\bf C})}\). We compute the singular locus of the family of commutative spherical symplectic reflection algebras deforming C4/G. We also discuss preliminary investigations on the more general question of classifying linear quotients V/G admitting symplectic resolutions.

Journal ArticleDOI
TL;DR: In this article, it was shown that a closed, oriented, connected three-manifold is dominated by a product if and only if it is finitely covered either by the product or by a connected sum of copies of S 2 × S 1.
Abstract: We determine which three-manifolds are dominated by products. The result is that a closed, oriented, connected three-manifold is dominated by a product if and only if it is finitely covered either by a product or by a connected sum of copies of S 2 × S 1. This characterization can also be formulated in terms of Thurston geometries, or in terms of purely algebraic properties of the fundamental group. We also determine which three-manifolds are dominated by non-trivial circle bundles, and which three-manifold groups are presentable by products.

Journal ArticleDOI
TL;DR: In this paper, the authors generalize Colding-Minicozzi's recent results about codimension-1 self-shrinkers for the mean curvature flow to higher codimensions.
Abstract: In this paper, we generalize Colding–Minicozzi’s recent results about codimension-1 self-shrinkers for the mean curvature flow to higher codimension. In particular, we prove that the sphere $${bf S}^{n}(\sqrt{2n})$$ is the only complete embedded connected $$F$$ -stable self-shrinker in $$\mathbf{R}^{n+k}$$ with $$\mathbf{H} e 0$$ , polynomial volume growth, flat normal bundle and bounded geometry. We also discuss some properties of symplectic self-shrinkers, proving that any complete symplectic self-shrinker in $$\mathbf{R}^4$$ with polynomial volume growth and bounded second fundamental form is a plane. As a corollary, we show that there is no finite time Type I singularity for symplectic mean curvature flow, which has been proved by Chen–Li using different method. We also study Lagrangian self-shrinkers and prove that for Lagrangian mean curvature flow, the blow-up limit of the singularity may be not $$F$$ -stable.

Journal ArticleDOI
TL;DR: In this article, the relation between self extensions of simple representations of quantum affine algebras and the property of a simple representation being prime was explored and a large class of prime representations satisfying this homological property was presented.
Abstract: We explore the relation between self extensions of simple representations of quantum affine algebras and the property of a simple representation being prime. We show that every nontrivial simple representation has a nontrivial self extension. Conversely, we prove that if a simple representation has a unique nontrivial self extension up to isomorphism, then its Drinfeld polynomial is a power of the Drinfeld polynomial of a prime representation. It turns out that, in the $$\mathfrak{sl }_2$$ -case, a simple module is prime if and only if it has a unique nontrivial self extension up to isomorphism. It is tempting to conjecture that this is true in general and we present a large class of prime representations satisfying this homological property.

Journal ArticleDOI
TL;DR: In this paper, it was shown that the set of Collet-Eckmann maps has positive Lebesgue measure in the space of rational maps on the Riemann sphere for any fixed degree d ≥ 2.
Abstract: We show that the set of Collet–Eckmann maps has positive Lebesgue measure in the space of rational maps on the Riemann sphere for any fixed degree d ≥ 2.

Journal ArticleDOI
TL;DR: In this paper, it was shown that the dynamics of a homeomorphism is annular, in the sense that there exists a periodic, essential, annular set for it.
Abstract: We prove that for a homeomorphism $$\tilde{f}:\mathbf T ^2\rightarrow \mathbf T ^2$$ in the homotopy class of the identity and with a lift $$f:\mathbf R ^2\rightarrow \mathbf R ^2$$ whose rotation set $$\rho (f)$$ is an interval, either every rational point in $$\rho (f)$$ is realized by a periodic orbit, or the dynamics of $$\tilde{f}$$ is annular, in the sense that there exists a periodic, essential, annular set for $$\tilde{f}$$ . In the latter case we also give a qualitative description of the dynamics.

Journal ArticleDOI
TL;DR: In this article, an A'Campo type formula for the tame monodromy zeta function of a smooth and proper variety over a discretely valued field K is presented.
Abstract: We prove an A’Campo type formula for the tame monodromy zeta function of a smooth and proper variety over a discretely valued field K. As a first application, we relate the orders of the tame monodromy eigenvalues on the l-adic cohomology of a K-curve to the geometry of a relatively minimal sncd-model, and we show that the semi-stable reduction theorem and Saito’s criterion for cohomological tameness are immediate consequences of this result. As a second application, we compute the error term in the trace formula for smooth and proper K-varieties. We see that the validity of the trace formula would imply a partial generalization of Saito’s criterion to arbitrary dimension.

Journal ArticleDOI
TL;DR: In this article, it was shown that W(ψ) is of full Lebesgue measure if there exists an ϵ > 0 such that ϵ ≥ ϵ.
Abstract: Given a nonnegative function \({\psi\mathbb{N} \to \mathbb{R}}\), let W(ψ) denote the set of real numbers x such that |nx − a| 0). A consequence of our main result is that W(ψ) is of full Lebesgue measure if there exists an \({\epsilon > 0}\) such that $$ \textstyle \sum_{n\in\mathbb{N}}\left(\frac{\psi(n)}{n}\right)^{1+\epsilon}\varphi (n)=\infty. $$ The Duffin–Schaeffer Conjecture is the corresponding statement with \({\epsilon = 0}\) and represents a fundamental unsolved problem in metric number theory. Another consequence is that W(ψ) is of full Hausdorff dimension if the above sum with \({\epsilon = 0}\) diverges; i.e. the dimension analogue of the Duffin–Schaeffer Conjecture is true.

Journal ArticleDOI
TL;DR: Goers et al. as discussed by the authors used the approach introduced in (Goerss et al., Ann Math 162(2):777-822, 2005) in order to analyze the homotopy groups of the mod-3 Moore spectrum.
Abstract: In this paper we use the approach introduced in (Goerss et al., Ann Math 162(2):777–822, 2005) in order to analyze the homotopy groups of \(L_{K(2)}V(0)\), the mod-\(3\) Moore spectrum \(V(0)\) localized with respect to Morava \(K\)-theory \(K(2)\). These homotopy groups have already been calculated by Shimomura (J Math Soc Japan 52(1): 65–90, 2000). The results are very complicated so that an independent verification via an alternative approach is of interest. In fact, we end up with a result which is more precise and also differs in some of its details from that of Shimomura (J Math Soc Japan 52(1): 65–90, 2000). An additional bonus of our approach is that it breaks up the result into smaller and more digestible chunks which are related to the \(K(2)\)-localization of the spectrum \(TMF\) of topological modular forms and related spectra. Even more, the Adams–Novikov differentials for \(L_{K(2)}V(0)\) can be read off from those for \(TMF\).