Authors:Pavel Andreev Abstract: Abstract We present an approach leading to Finsler geometry without differential calculus of tensors. Several natural examples of such singular Finsler spaces are studied. One class of such examples contains Busemann G-spaces with non-positive curvature. Starting with a singular version of the axiomatics, some simplest properties known in the smooth Finsler geometry are interpreted. PubDate: 2017-08-07 DOI: 10.1007/s40879-017-0169-x

Authors:Patricio Gallardo; Evangelos Routis Abstract: Abstract We introduce and study smooth compactifications of the moduli space of n labeled points with weights in projective space, which have normal crossings boundary and are defined as GIT quotients of the weighted Fulton–MacPherson compactification. We show that the GIT quotient of a wonderful compactification is also a wonderful compactification under certain hypotheses. We also study a weighted version of the configuration spaces parametrizing n points in affine space up to translation and homothety. In dimension one, the above compactifications are isomorphic to Hassett’s moduli space of rational weighted stable curves. PubDate: 2017-08-07 DOI: 10.1007/s40879-017-0170-4

Authors:Jeffrey Yelton Abstract: Abstract Let K be a field of characteristic different from 2 and let E be an elliptic curve over K, defined either by an equation of the form \(y^{2} = f(x)\) with degree 3 or as the Jacobian of a curve defined by an equation of the form \(y^{2} = f(x)\) with degree 4. We obtain generators over K of the 8-division field K(E[8]) of E given as formulas in terms of the roots of the polynomial f, and we explicitly describe the action of a particular automorphism in \(\mathrm {Gal}(K(E[8]) / K)\) . PubDate: 2017-08-01 DOI: 10.1007/s40879-017-0162-4

Authors:Valerii N. Berestovskii Abstract: Abstract The author proved in the late 1980s that any homogeneous manifold with an intrinsic metric is isometric to some homogeneous quotient space of a connected Lie group by its compact subgroup with an invariant Finslerian or sub-Finslerian metric. In the case of a trivial compact subgroup, the invariant Riemannian or sub-Riemannian metrics are singled out from invariant Finslerian or sub-Finslerian metrics by being in one-to-one correspondence with special one-parameter Gaussian convolutions semigroups of absolutely continuous probability measures. Any such semigroup is generated by a second order hypoelliptic operator. In the present paper, in connection with this, the author discusses briefly the operator definition of lower bound for Ricci curvature by Baudoin–Garofalo. Earlier, Agrachev defined a notion of curvature for sub-Riemannian manifolds. As an alternative, the author discusses in some detail old definitions of curvature tensors for rigged metrized distributions on manifolds given by Schouten, Wagner, and Solov’ev. To calculate the Solov’ev sectional and Ricci curvatures for homogeneous sub-Riemannian manifolds, the author suggests to use in some cases special riggings of invariant bracket generating distributions on manifolds. As a justification, we find a foliation on the cotangent bundle over a Lie group G whose leaves are tangent to invariant Hamiltonian vector fields for the Pontryagin–Hamilton function. This function was applied in the Pontryagin maximum principle for the time-optimal problem. The foliation is entirely described by the coadjoint representation of the Lie group G. We also use the canonical symplectic form on and its values for the above mentioned invariant Hamiltonian vector fields. In particular, the above rigging method is applicable to contact sub-Riemannian manifolds, sub-Riemannian Carnot groups, and homogeneous sub-Riemannian manifolds possessing a submetry onto a Riemannian manifold. In Sects. 5 and 6, some examples are presented. PubDate: 2017-07-31 DOI: 10.1007/s40879-017-0171-3

Authors:Stephan Baier Abstract: Abstract We study the problem of Diophantine approximation on lines in \(\mathbb {C}^2\) with numerators and denominators restricted to Gaussian primes. To this end, we develop analogs of well-known results on small fractional parts of \(p\gamma , p\) running over the primes and \(\gamma \) being a fixed irrational, for Gaussian primes. PubDate: 2017-07-27 DOI: 10.1007/s40879-017-0165-1

Authors:Ruben A. Hidalgo Abstract: Abstract A closed Riemann surface of genus \(g \geqslant 2\) is called a Hurwitz curve if its group of conformal automorphisms has order \(84(g-1)\) . In 1895, Wiman noticed that there is no Hurwitz curve of genus \(g=2,4,5,6\) and, up to isomorphisms, there is a unique Hurwitz curve of genus \(g=3\) ; this being Klein’s plane quartic curve. Later, in 1965, Macbeath proved the existence, up to isomorphisms, of a unique Hurwitz curve of genus \(g=7\) ; this known as the Fricke–Macbeath curve. Equations were also provided; that being the fiber product of suitable three elliptic curves. In the same year, Edge constructed such a genus seven Hurwitz curve by elementary projective geometry. Such a construction was provided by first constructing a 4-dimensional family of genus seven closed Riemann surfaces \(S_{\mu }\) admitting a group of conformal automorphisms so that \(S_{\mu }/G_{\mu }\) has genus zero. In this paper we discuss the above curves in terms of fiber products of classical Fermat curves and we provide a geometrical explanation of the three elliptic curves in Macbeath’s description. We also observe that the jacobian variety of each \(S_{\mu }\) is isogenous to the product of seven elliptic curves (explicitly given) and, for the particular Fricke–Macbeath curve, we obtain the well-known fact that its jacobian variety is isogenous to \(E^{7}\) for a suitable elliptic curve E. PubDate: 2017-07-24 DOI: 10.1007/s40879-017-0166-0

Authors:William Chen Abstract: Abstract We solve several problems about different variants of the principle. First, we separate principles which prescribe certain order-types for the members of the family. Then, we separate a principle called \(\mathsf {Superstick}\) from \(\mathsf {CH}\) , answering a question of Primavesi. PubDate: 2017-07-20 DOI: 10.1007/s40879-017-0167-z

Authors:Cristian D. González-Avilés Abstract: Abstract Let S be a reduced scheme and let \(f:X\rightarrow S\) and \(g:Y\rightarrow S\) be faithfully flat morphisms locally of finite presentation with reduced and connected maximal geometric fibers. We discuss the canonical maps and induced by f and g. Under certain additional conditions on S, X, Y, f and g, we describe the kernel and cokernel of the preceding maps, thereby extending known results when S is the spectrum of a field. PubDate: 2017-07-12 DOI: 10.1007/s40879-017-0164-2

Authors:Weixu Su; Youliang Zhong Abstract: Abstract This paper is a brief survey of some results on the Finsler structures of Teichmüller metric. We mainly describe the connection between holomorphic quadratic differentials and extremal length of measured foliations. Some rigidity results about the Teichmüller metric inspired by Royden’s theorem are discussed. Teichmüller theory is probably the best example of an instance where Finsler geometry enters in the field of low dimensional geometry and topology. PubDate: 2017-07-11 DOI: 10.1007/s40879-017-0161-5

Authors:Sven Pistre; Heiko von der Mosel Abstract: Abstract We investigate a connection between the two-dimensional Finslerian area functional in arbitrary codimension based on the Busemann–Hausdorff volume form, and well-investigated Cartan functionals to solve the Plateau problem in Finsler spaces. This generalises a previously known result due to Overath and von der Mosel (Manuscripta Math 143(3–4):273–316, 2014) to higher codimension. PubDate: 2017-07-05 DOI: 10.1007/s40879-017-0163-3

Authors:Gennadii S. Asanov Abstract: Abstract A pseudo-Finsleroid metric function F of the two-axes structure that involves the vertical axis and the horizontal axis is proposed assuming constancy of the curvature of indicatrix. The curvature is negative and the signature of the Finslerian metric tensor is exactly \((+-\cdots )\) . The function F endows the tangent space with the geometry which possesses many interesting Finslerian properties. The use of the angle representation is the underlying method which has been conveniently and successfully applied. The appearance of the positive-definite Finsleroid metric function in the horizontal sections of the tangent space is established. PubDate: 2017-06-30 DOI: 10.1007/s40879-017-0160-6

Authors:Ari Shnidman Abstract: Abstract We compute the Grothendieck group of the category of abelian varieties over an algebraically closed field k. We also compute the Grothendieck group of the category of A-isotypic abelian varieties, for any simple abelian variety A, assuming k has characteristic 0, and for any elliptic curve A in any characteristic. PubDate: 2017-06-27 DOI: 10.1007/s40879-017-0159-z

Authors:Max Dörner; Hansjörg Geiges; Kai Zehmisch Abstract: Abstract We survey some results on the existence (and non-existence) of periodic Reeb orbits on contact manifolds, both in the open and closed case. We place these statements in the context of Finsler geometry by including a proof of the folklore theorem that the Finsler geodesic flow can be interpreted as a Reeb flow. As a mild extension of previous results we present existence statements on periodic Reeb orbits on contact manifolds with suitable supporting open books. PubDate: 2017-06-22 DOI: 10.1007/s40879-017-0158-0

Authors:Thomas Creutzig Abstract: Abstract The Schur index of the \((A_1, X_n)\) -Argyres–Douglas theory is conjecturally a character of a vertex operator algebra. Here such vertex algebras are found for the \(A_{\text {odd}}\) and \(D_{\text {even}}\) -type Argyres–Douglas theories. The vertex operator algebra corresponding to \(A_{2p-3}\) -Argyres–Douglas theory is the logarithmic -algebra of Creutzig et al. (Lett Math Phys 104(5):553–583, 2014), while the one corresponding to \(D_{2p}\) , denoted by , is realized as a non-regular quantum Hamiltonian reduction of \(L_{k}(\mathfrak {sl}_{p+1})\) at level . For all n one observes that the quantum Hamiltonian reduction of the vertex operator algebra of \(D_n\) -Argyres–Douglas theory is the vertex operator algebra of \(A_{n-3}\) -Argyres–Douglas theory. As a corollary, one realizes the singlet and triplet algebras (the vertex algebras associated to the best understood logarithmic conformal field theories) as quantum Hamiltonian reductions as well. Finally, characters of certain modules of these vertex operator algebras and the modular properties of their meromorphic continuations are given. PubDate: 2017-06-22 DOI: 10.1007/s40879-017-0156-2

Authors:Libing Huang Abstract: Abstract We introduce the notions of spray vector and connection operator to give efficient curvature formulas for a homogeneous Finsler space. Thus the flag curvatures can be computed in the Lie algebra level. Applying these formulas, one can show that in several occasions the structure of the Lie algebra may have influence over the signs of the flag curvatures, regardless of the underlying Finsler metric. Some concrete examples are constructed to illustrate the concepts and the curvature behavior in Finsler geometry. PubDate: 2017-06-22 DOI: 10.1007/s40879-017-0157-1

Authors:Michael McQuillan Abstract: Abstract The classification of foliated surfaces (McQuillan in Pure Appl Math Q 4(3):877–1012, 2008) is applied to the study of curves on surfaces with big co-tangent bundle and varying moduli, be it purely in characteristic zero, or, more generally when the characteristic is mixed. Almost everything that one might naively imagine is true, but with one critical exception: rational curves on bi-disc quotients which aren’t quotients of products of curves are Zariski dense in mixed characteristic. The logical repercussions in characteristic zero of this exception are not negligible. PubDate: 2017-06-12 DOI: 10.1007/s40879-017-0141-9

Authors:Nefton Pali Abstract: Abstract The concavity of Perelman’s \(\mathcal {W}\) -functional over a neighborhood of a Kähler–Ricci soliton inside the space of Kähler potentials is a direct consequence of author’s solution of the variational stability problem for Kähler–Ricci solitons. We provide a new and rather simple proof of this particular fact. This new proof uses in minor part some elementary formulas obtained in our previous work. PubDate: 2017-06-08 DOI: 10.1007/s40879-017-0155-3

Authors:Csaba Vincze Abstract: Abstract A generalized Berwald manifold is a Finsler manifold admitting a linear connection on the base manifold such that parallel transports preserve the Finslerian length of tangent vectors (compatibility condition). If the linear connection preserving the Finslerian length of tangent vectors has zero torsion then we have a classical Berwald manifold. Another important type of generalized Berwald manifolds admits semi-symmetric compatible linear connections. This means that the torsion tensor is decomposable in a special way. Among others the so-called Wagner manifolds belong to this special class of spaces. In the sense of Hashiguchi and Ychijyo’s classical results, Wagner manifolds play an important role in the conformal Finsler geometry as conformally Berwald Finsler manifolds. One of the main problems is the intrinsic characterization of compatible linear connections on a Finsler manifold. We are also interested in the inverse problem of compatible linear connections on Finsler manifolds. Let a metric linear connection \(\nabla \) on a Riemannian manifold be given. After formulating a necessary and sufficient condition for \(\nabla \) to be metrizable by a non-Riemannian metric function we present a geometric construction of the (non-Riemannian) indicatrix hypersurfaces in terms of generalized conics—in case of a Riemannian manifold the indicatrices are conics (quadratic hypersurfaces) in the classical sense. New perspectives of the theory of generalized Berwald manifolds have been supported by the solution of Matsumoto’s problem of conformally equivalent Berwald manifolds in 2005: the scale function between two non-Riemannian (classical) Berwald manifolds must be constant. The proof is based on metrics and differential forms given by averaging. They also play the central role in the intrinsic characterization of semi-symmetric compatible linear connections in general. Using average processes is a new and important trend in Finsler geometry. The so-called associated Riemannian metric is introduced by choosing the Riemann–Finsler metric to be averaged. Another important type of associated objects is the so-called associated Randers metric. The one-form perturbation of the associated Riemannian metric is given by the integration of the contracted-normalized Riemann–Finsler metric on the indicatrix hypersurface point by point. Some applications are also presented in case of Funk metrics. Since the associated objects inherit the compatibility properties, the generalized Berwald manifold theory for spaces of special metrics is of special interest. In what follows we present some recent results of the theory of generalized Berwald manifolds. Especially we focus on the case of semi-symmetric compatible linear connections. We also discuss the case of Randers metrics. Asanov’s Finsleroid-Finsler metrics will be characterized as the solutions of a conformal rigidity problem and we prove that a Finsleroid-Finsler manifold is a Landsberg manifold (Unicorn) if and only if it is a generalized Berwald manifold with a semi-symmetric compatible linear connection. PubDate: 2017-06-08 DOI: 10.1007/s40879-017-0153-5

Authors:Evgeny Shinder Abstract: Abstract After recalling basic definitions and constructions for a finite group G action on a k-linear category we give a concise proof of the following theorem of Elagin: if \(\mathscr {C}= \langle \mathscr {A}, \mathscr {B}\rangle \) is a semiorthogonal decomposition of a triangulated category which is preserved by the action of G, and \(\mathscr {C}^G\) is triangulated, then there is a semiorthogonal decomposition \(\mathscr {C}^G = \langle \mathscr {A}^G, \mathscr {B}^G \rangle \) . We also prove that any G-action on \(\mathscr {C}\) is weakly equivalent to a strict G-action which is the analog of the Coherence theorem for monoidal categories. PubDate: 2017-06-02 DOI: 10.1007/s40879-017-0150-8

Authors:Piotr Jędrzejewicz; Janusz Zieliński Abstract: Abstract We present some motivations and discuss various aspects of an approach to the Jacobian Conjecture in terms of irreducible elements and square-free elements. PubDate: 2017-05-01 DOI: 10.1007/s40879-017-0145-5