Showing papers on "Complex dimension published in 2015"

Topological Dimension and Dynamical Systems

Abstract: Translated from the popular French edition, the goal of the book is to provide a self-contained introduction to mean topological dimension, an invariant of dynamical systems introduced in 1999 by Misha Gromov. The book examines how this invariant was successfully used by Elon Lindenstrauss and Benjamin Weiss to answer a long-standing open question about embeddings of minimal dynamical systems into shifts. A large number of revisions and additions have been made to the original text. Chapter 5 contains an entirely new section devoted to the Sorgenfrey line. Two chapters have also been added: Chapter 9 on amenable groups and Chapter 10 on mean topological dimension for continuous actions of countable amenable groups. These new chapters contain material that have never before appeared in textbook form. The chapter on amenable groups is based on Folner s characterization of amenability and may be read independently from the rest of the book. Although the contents of this book lead directly to several active areas of current research in mathematics and mathematical physics, the prerequisites needed for reading it remain modest; essentially some familiarities with undergraduate point-set topology and, in order to access the final two chapters, some acquaintance with basic notions in group theory. Topological Dimension and Dynamical Systems is intended for graduate students, as well as researchers interested in topology and dynamical systems. Some of the topics treated in the book directly lead to research areas that remain to be explored

Lipschitz regular complex algebraic sets are smooth

Abstract: A classical Theorem of Mumford implies that a topologically regular complex algebraic surface in C 3 with an isolated singular point is smooth. We prove that any Lipschitz regular complex algebraic set is smooth. No restriction on the dimension and no restriction on the singularity to be isolated is needed.

Quantization of open-closed BCOV theory, I

Abstract: This is the first in a series of papers which analyze the problem of quantizing the theory coupling Kodaira-Spencer gravity (or BCOV theory) on Calabi-Yau manifolds using the formalism for perturbative QFT developed by the first author. In this paper, we focus on flat space $\mathbb{C}^d$ for $d$ odd. We prove that there exists a unique quantization of the theory coupling BCOV theory and holomorphic Chern-Simons theory with gauge group the supergroup $GL(N \mid N)$. We deduce a canonically defined quantization of BCOV theory on its own. We also discuss some conjectural links between BCOV theory in various dimensions and twists of physical theories: in complex dimension $3$ we conjecture a relationship to twists of $(1,0)$ supersymmetric theories and in complex dimension $5$ to a twist of type IIB supergravity.

System and method for generating codebooks with small projections per complex dimension and utilization thereof

Abstract: A method for data transmission by a device in a communication system includes modulating a first data stream using a codebook to produce a second data stream, wherein the codebook is in correspondence with a multi-dimensional modulation map that includes a number of distinct projections per complex dimension that is smaller than a number of modulation points of the multi-dimensional modulation map, and transmitting the second data stream over allocated resources in the communication system.

Estimating singularity dimension

Abstract: In this paper we address an interesting question on the computation of the dimension of self-affine sets in Euclidean space. A well-known result of Falconer showed that under mild assumptions the Hausdorff dimension of typical self-affine sets is equal to its Singularity dimension. Heuter and Lalley subsequently presented a smaller open family of non-trivial examples for which there is an equality of these two dimensions. In this article we analyse the size of this family and present an efficient algorithm for estimating the dimension.

Probabilistic Algorithm for Computing the Dimension of Real Algebraic Sets

Abstract: Let fE Q[X1, …, Xn] be a polynomial of degree D. We consider the problem of computing the real dimension of the real algebraic set defined by f=0. Such a problem can be reduced to quantifier elimination. Hence it can be tackled with Cylindrical Algebraic Decomposition within a complexity that is doubly exponential in the number of variables. More recently, denoting by d the dimension of the real algebraic set under study, deterministic algorithms running in time DO(d(n-d)) have been proposed. However, no implementation reflecting this complexity gain has been obtained and the constant in the exponent remains unspecified.We design a probabilistic algorithm which runs in time which is essentially cubic in Dd(n-d). Our algorithm takes advantage of genericity properties of polar varieties to avoid computationally difficult steps of quantifier elimination. We also report on a first implementation. It tackles examples that are out of reach of the state-of-the-art and its practical behavior reflects the complexity gain.

Spectral Dimension of kappa-deformed space-time

Abstract: We investigate the spectral dimension of $\kappa$-space-time using the $\kappa$-deformed diffusion equation. The deformed equation is constructed for two different choices of Laplacians in $n$-dimensional, $\kappa$-deformed Euclidean space-time. We use an approach where the deformed Laplacians are expressed in the commutative space-time itself. Using the perturbative solutions to diffusion equations, we calculate the spectral dimension of $\kappa$-deformed space-time and show that it decreases as the probe length decreases. By introducing a bound on the deformation parameter, spectral dimension is guaranteed to be positive definite. We find that, for one of the choices of the Laplacian, the non-commutative correction to the spectral dimension depends on the topological dimension of the space-time whereas for the other, it is independent of the topological dimension. We have also analysed the dimensional flow for the case where the probe particle has a finite extension, unlike a point particle.

Regularity scales and convergence of the Calabi flow

Abstract: We define regularity scales to study the behavior of the Calabi flow. Based on estimates of the regularity scales, we obtain convergence theorems of the Calabi flow on extremal Kahler surfaces, under the assumption of global existence of the Calabi flow solutions. Our results partially confirm Donaldson's conjectural picture for the Calabi flow in complex dimension 2. Similar results hold in high dimension with an extra assumption that the scalar curvature is uniformly bounded.

A computing procedure for the small inductive dimension of a finite \mathrm{T}_0-space

Abstract: The most important properties of small inductive dimension (ind) are well known (see, for example, Engelking in Theory of dimensions, finite and infinite, 1995 and Pears in Dimension theory of general spaces, 1975). In this paper, we characterize this dimension of a finite \(\mathrm{T}_0\)-space using matrix algebra. Therefore, using this characterization, we present an algorithm for computing the dimension ind and we compute an upper bound on the number of iterations of the algorithm. Finally, some remarks and open questions are posed.

Families of Riemann Surfaces, Uniformization and Arithmeticity

Abstract: A consequence of the results of Bers and Griffiths on the uniformization of complex algebraic varieties is that the universal cover of a family of Riemann surfaces, with base and fibers of finite hyperbolic type, is a contractible 2-dimensional domain that can be realized as the graph of a holomorphic motion of the unit disk. In this paper we determine which holomorphic motions give rise to these uniformizing domains and characterize which among them correspond to arithmetic families (i.e. families defined over number fields). Then we apply these results to characterize the arithmeticity of complex surfaces of general type in terms of the biholomorphism class of the 2-dimensional domains that arise as universal covers of their Zariski open subsets. For the important class of Kodaira fibrations this criterion implies that arithmeticity can be read off from the universal cover. All this is very much in contrast with the corresponding situation in complex dimension one, where the universal cover is always the unit disk.

Spectral Action Models of Gravity on Packed Swiss Cheese Cosmology

Abstract: We present a model of (modified) gravity on spacetimes with fractal structure based on packing of spheres, which are (Euclidean) variants of the Packed Swiss Cheese Cosmology models. As the action functional for gravity we consider the spectral action of noncommutative geometry, and we compute its expansion on a space obtained as an Apollonian packing of 3-dimensional spheres inside a 4-dimensional ball. Using information from the zeta function of the Dirac operator of the spectral triple, we compute the leading terms in the asymptotic expansion of the spectral action. They consist of a zeta regularization of a divergent sum which involves the leading terms of the spectral actions of the individual spheres in the packing. This accounts for the contribution of the points 1 and 3 in the dimension spectrum (as in the case of a 3-sphere). There is an additional term coming from the residue at the additional point in the real dimension spectrum that corresponds to the packing constant, as well as a series of fluctuations coming from log-periodic oscillations, created by the points of the dimension spectrum that are off the real line. These terms detect the fractality of the residue set of the sphere packing. We show that the presence of fractality influences the shape of the slow-roll potential for inflation, obtained from the spectral action. We also discuss the effect of truncating the fractal structure at a certain scale related to the energy scale in the spectral action.

A bound on the second Betti number of hyperk\"ahler manifolds of complex dimension six

Abstract: Let M be an irreducible compact hyperkahler manifold of complex dimension six. We prove that the second Betti number of M is at most 23.

Sharp value for the Hausdorff dimension of the range and the graph of stable-like processes

Abstract: We determine the Hausdorff dimension for the range of a class of pure jump Markov processes in $\mathbb{R}^d$, which turns out to be random and depends on the trajectories of these processes. The key argument is carried out through the SDE representation of these processes. The method developed here also allows to compute the Hausdorff dimension for the graph.

PI spaces with analytic dimension 1 and arbitrary topological dimension

Abstract: For every n, we construct a metric measure space that is doubling, satisfies a Poincare inequality in the sense of Heinonen-Koskela, has topological dimension n, and has a measurable tangent bundle of dimension 1.

Isolated singularities of affine special K\"ahler metrics in two dimensions

Abstract: We prove that there are just two types of isolated singularities of special Kahler metrics in real dimension two provided the associated holomorphic cubic form does not have essential singularities. We also construct examples of such metrics.

A Simple Example Concerning the Upper Box-Counting Dimension of a Cartesian Product

Abstract: We give a simple example of two countable sets X and Y of real numbers such that their upper box-counting dimension satisfies the strict inequality dimb(X × Y ) < dimb(X) + dimb(Y ).

Log Kodaira dimension of homogeneous varieties

Abstract: Let $V$ be a complex algebraic variety, homogeneous under the action of a complex algebraic group. We show that the log Kodaira dimension of $V$ is non-negative if and only if $V$ is a semi-abelian variety.

Compactifications of moduli spaces and cellular decompositions

Abstract: By a Riemann surface or simply a curve we mean a compact connected complex manifold of complex dimension one. Denote by Mg;n the moduli space of Riemann surfaces with genus g and n> 0 labeled points. The Deligne–Mumford compactification is denoted by Mg;n . This is a space parametrizing stable Riemann surfaces. Here the word “stable” refers to the finiteness of the group of conformal automorphisms of the surface. Geometrically it means that we only allow double point (also called node) singularities and that each irreducible component of the surface has negative Euler characteristic (taking the labeled points and nodes into account). We can further perform a real oriented blowup along the locus of degenerate surfaces to obtain the space Mg;n . Intuitively, this space is similar to the Deligne–Mumford space but it also remembers the angle at each double point at which the surface degenerated. The decorated moduli space is denoted by Mdec g;n DMg;n  1 , where  1 is the .n 1/–dimensional standard simplex. The decorations can be thought of as hyperbolic lengths of certain horocycles or as quadratic residues of Strebel–Jenkins differentials on a Riemann surface. By choosing an appropriate notion of decoration on a stable Riemann Surface it is possible to construct compactifications Mdec g;n and Mdec g;n . The first main result of this paper identifies the homeomorphism type of these compactifications. Let P be a finite set of labels.

Instanton calculus without equations of motion: Semiclassics from monodromies of a Riemann surface

Abstract: Instanton calculations in semiclassical quantum mechanics rely on integration along trajectories which solve classical equations of motion. However in systems with higher dimensionality or complexified phase space these are rarely attainable. A prime example are spin-coherent states which are used e.g. to describe single molecule magnets (SMM). We use this example to develop instanton calculus which does not rely on explicit solutions of the classical equations of motion. Energy conservation restricts the complex phase space to a Riemann surface of complex dimension one, allowing to deform integration paths according to Cauchy's integral theorem. As a result, the semiclassical actions can be evaluated without knowing actual classical paths. Furthermore we show that in many cases such actions may be solely derived from monodromy properties of the corresponding Riemann surface and residue values at its singular points. As an example, we consider quenching of tunneling processes in SMM by an applied magnetic field.

1-complete semiholomorphic foliations

Abstract: A semiholomorphic foliations of type (n, d) is a differentiable real manifold X of dimension 2n + d, foliated by complex leaves of complex dimension n. In the present work, we introduce an appropriate notion of pseudoconvexity (and consequently, q-completeness) for such spaces, given by the interplay of the usual pseudoconvexity, along the leaves, and the positivity of the transversal bundle. For 1-complete real analytic semiholomorphic foliations, we obtain a vanishing theorem for the CR cohomology, which we use to show an extension result for CR functions on Levi flat hypersurfaces and an embedding theorem in C^N . In the compact case, we introduce a notion of weak positivity for the transversal bundle, which allows us to construct a real analytic embedding in CP^N .

Dynamics of Polynomial Diffeomorphisms of $C^2$: Foliations and laminations

Abstract: This essay summarizes the state of the art on some aspects of the dynamics of polynomial diffeomorphsms in complex dimension two, and it presents a number of open questions.

Examples of simply-connected K-contact non-Sasakian manifolds of dimension 5

Abstract: The existence of compact simply-connected K-contact, but not Sasakian, manifolds has been unknown only for dimension 5. The aim of this paper is to show that the Kollar's simply-connected example which is a Seifert bundle over the complex projective space ℂℙ2 and does not carry any Sasakian structure is actually a K-contact manifold. As a consequence, we affirmatively answer the above existence problem in dimension 5, establishing that there are infinitely many compact simply-connected K-contact manifolds of dimension 5 which do not carry a Sasakian structure.

Isolated Singularities of Affine Special Kähler Metrics in Two Dimensions

Abstract: We prove that there are just two types of isolated singularities of special Kahler metrics in real dimension two, provided the associated holomorphic cubic form does not have essential singularities. We also construct examples of such metrics.

Embedding Factors for Branching in Hermitian Clifford Analysis

Abstract: A step 2 branching decomposition of spaces of homogeneous Hermitian monogenic polynomials in \({\mathbb {C}}^n\) is established with explicit embedding factors in terms of the generalized Jacobi polynomials, which allows for an inductive construction of an orthogonal basis for those spaces The embedding factors and the orthogonal bases are fully worked out in the complex dimension 2 case, with special interest for the Appell property

Submaximally symmetric c-projective structures

Abstract: C-projective structures are analogues of projective structures in the complex setting. The maximal dimension of the Lie algebra of c-projective symmetries of a complex connection on an almost complex manifold of C-dimension $n>1$ is classically known to be $2n^2+4n$. We prove that the submaximal dimension is equal to $2n^2-2n+4+2\delta_{3,n}$. If the complex connection is minimal (encoded as a normal parabolic geometry), the harmonic curvature of the c-projective structure has three components and we specify the submaximal symmetry dimensions and the corresponding geometric models for each of these three pure curvature types. If the connection is non-minimal, we introduce a modified normalization condition on the parabolic geometry and use this to resolve the symmetry gap problem. We prove that the submaximal symmetry dimension in the class of Levi-Civita connections for pseudo-K\"ahler metrics is $2n^2-2n+4$, and specializing to the K\"ahler case, we obtain $2n^2-2n+3$. This resolves the symmetry gap problem for metrizable c-projective structures.

Affine connections on complex manifolds of algebraic dimension zero

Abstract: We prove that any compact complex manifold with finite fundamental group and algebraic dimension zero admits no holomorphic affine connection.

Geometric dimension of lattices in classical simple Lie groups

Abstract: We prove that if $\Gamma$ is a lattice in a classical simple Lie group $G$, then the symmetric space of $G$ is $\Gamma$-equivariantly homotopy equivalent to a proper cocompact $\Gamma$-CW complex of dimension the virtual cohomological dimension of $\Gamma$.