Authors:Igor Burban; Yuriy Drozd; Volodymyr Gavran Pages: 311 - 341 Abstract: We develop the theory of minors of non-commutative schemes. This study is motivated by applications in the theory of non-commutative resolutions of singularities of commutative schemes. In particular, we construct a categorical resolution for non-commutative curves and in the rational case show that it can be realized as the derived category of a quasi-hereditary algebra. PubDate: 2017-06-01 DOI: 10.1007/s40879-017-0128-6 Issue No:Vol. 3, No. 2 (2017)

Authors:Matthias Kunik Pages: 363 - 378 Abstract: The Farey sequence of order n consists of all reduced fractions a / b between 0 and 1 with positive denominator b less or equal to n. The sums of the inverse denominators 1 / b of the Farey fractions in prescribed intervals with rational bounds have simple main terms, whereas the deviations are determined by a sequence of polygonal functions \(f_n\) . In a former paper we obtained a limit function for \(n \rightarrow \infty \) which describes an asymptotic scaling property of functions \(f_n\) in the vicinity of any fixed fraction a / b and which is independent of a / b. In this paper we derive new representation formulas for \(f_n\) and related functions which give much better remainder term estimates. We also combine these results with those from our previous papers in order to prove that the sequence of functions \(f_n\) converges pointwise to zero. PubDate: 2017-06-01 DOI: 10.1007/s40879-017-0132-x Issue No:Vol. 3, No. 2 (2017)

Authors:Max Dörner; Hansjörg Geiges; Kai Zehmisch 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: 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: 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: 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: 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: 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: 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:Aleksandr V. Pukhlikov Abstract: It is proved that the global log canonical threshold of a Zariski general Fano complete intersection of index 1 and codimension k in \({\mathbb P}^{M+k}\) is equal to one, if \(M\geqslant 2k+3\) and the maximum of the degrees of defining equations is at least 8. This is an essential improvement of the previous results about log canonical thresholds of Fano complete intersections. As a corollary we obtain the existence of Kähler–Einstein metrics on generic Fano complete intersections described above. PubDate: 2017-05-30 DOI: 10.1007/s40879-017-0152-6

Authors:Dirk Siersma; Mihai Tibăr Abstract: We introduce and study the vanishing homology of singular projective hypersurfaces. We prove its concentration in two levels in case of 1-dimensional singular locus \({\Sigma }\) , and moreover determine the ranks of the nontrivial homology groups. These two groups depend on the monodromy at special points of \({\Sigma }\) and on the effect of the monodromy of the local system over its complement. PubDate: 2017-05-30 DOI: 10.1007/s40879-017-0151-7

Authors:Maria A. Bertolim; Dahisy V. S. Lima; Margarida P. Mello; Ketty A. de Rezende; Mariana R. da Silveira Abstract: We study algorithms that give rise to a global Smale’s Cancellation Theorem for dimensions \(n\geqslant 6\) . The Spectral Sequence Sweeping Algorithm (SSSA) and the Row Cancellation Algorithm (RCA) for a filtered Morse chain complex on a manifold \(M^{n}\) are presented. Our main theorems, which make use of these algorithms with a connection matrix as an input, establish a correspondence between algebraic cancellations in a spectral sequence and dynamical cancellations of the gradient flow on \(M^{n}\) for dimensions \(n\geqslant 6\) . PubDate: 2017-05-10 DOI: 10.1007/s40879-017-0144-6

Authors:Piotr Jędrzejewicz; Janusz Zieliński 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

Authors:Mira Bivas Abstract: The main result in this paper is an existence theorem for constraint differential inclusions whose right-hand side satisfies some Carathéodory semicontinuity assumptions of mixed type. It generalises all known theorems in the field in the finite-dimensional case. As preliminary results needed in the proof we obtain some interrelations between measurability and \(\varepsilon \) -semicontinuity of single- and set-valued mappings. PubDate: 2017-04-26 DOI: 10.1007/s40879-017-0142-8

Authors:Matt Kerr; Colleen Robles Abstract: We show that the smooth horizontal Schubert subvarieties of a rational homogeneous variety G / P are homogeneously embedded cominuscule , and are classified by subdiagrams of a Dynkin diagram. This generalizes the classification of smooth Schubert varieties in cominuscule G / P. PubDate: 2017-04-20 DOI: 10.1007/s40879-017-0140-x

Authors:Ryan R. Martin; Shanise Walker Abstract: The \(\mathscr {N}\) poset consists of four distinct sets W, X, Y, Z such that \(W\subset X, Y\subset X\) , and \(Y\subset Z\) where W is not necessarily a subset of Z. A family \({{\mathscr {F}}}\) , considered as a subposet of the n-dimensional Boolean lattice \(\mathscr {B}_n\) , is \(\mathscr {N}\) -free if it does not contain \(\mathscr {N}\) as a subposet. Let \(\mathrm{La}(n, \mathscr {N})\) be the size of a largest \(\mathscr {N}\) -free family in \(\mathscr {B}_n\) . Katona and Tarján proved that , where and is the size of a single-error-correcting code with constant weight \(k+1\) . In this note, we prove for n even and \(k=n/2, \mathrm{La}(n, \mathscr {N}) \geqslant {n\atopwithdelims ()k}+A(n, 4, k)\) , which improves the bound on \(\mathrm{La}(n, \mathscr {N})\) in the second order term for some values of n and should be an improvement for an infinite family of values of n, depending on the behavior of the function . PubDate: 2017-04-06 DOI: 10.1007/s40879-017-0139-3

Authors:Maxime Fairon Abstract: This paper aims at setting out the basics of \(\mathbb {Z}\) -graded manifolds theory. We introduce \(\mathbb {Z}\) -graded manifolds from local models and give some of their properties. The requirement to work with a completed graded symmetric algebra to define functions is made clear. Moreover, we define vector fields and exhibit their graded local basis. The paper also reviews some correspondences between differential \(\mathbb {Z}\) -graded manifolds and algebraic structures. PubDate: 2017-03-27 DOI: 10.1007/s40879-017-0138-4

Authors:Jan-Hendrik de Wiljes Abstract: We introduce the uniform coprime hypergraph of integers \(\mathrm{CHI}_k\) which is the graph with vertex set \({\mathbb {Z}}\) and a -hyperedge exactly between every \(k+1\) elements of \({\mathbb {Z}}\) having greatest common divisor equal to 1. This generalizes the concept of coprime graphs. We obtain some basic properties of these graphs and give upper and lower bounds for the clique number of certain subgraphs of \(\mathrm{CHI}_k\) . PubDate: 2017-03-15 DOI: 10.1007/s40879-017-0137-5

Authors:Emily Cliff Abstract: We introduce stacks classifying étale germs of pointed n-dimensional varieties. We show that quasi-coherent sheaves on these stacks are universal \(\mathscr {D}\) - and \(\mathscr {O}\) -modules. We state and prove a relative version of Artin’s approximation theorem, and as a consequence identify our stacks with classifying stacks of automorphism groups of the n-dimensional formal disc. We introduce the notion of convergent universal modules, and study them in terms of these stacks and the representation theory of the automorphism groups. PubDate: 2017-03-09 DOI: 10.1007/s40879-017-0135-7

Authors:Dino Lorenzini Abstract: Néron models were introduced by André Néron in his seminar at the IHÉS in 1961. This article gives a brief survey of past and recent results in this very useful theory. PubDate: 2017-03-01 DOI: 10.1007/s40879-017-0134-8