Showing papers on "Complex dimension published in 2017"

Operator Dimensions from Moduli

Abstract: We consider the operator spectrum of a three-dimensional $$ \mathcal{N}=2 $$ superconformal field theory with a moduli space of one complex dimension, such as the fixed point theory with three chiral superfields X, Y, Z and a superpotential W = XYZ. By using the existence of an effective theory on each branch of moduli space, we calculate the anomalous dimensions of certain low-lying operators carrying large R-charge J. While the lowest primary operator is a BPS scalar primary, the second-lowest scalar primary is in a semi-short representation, with dimension exactly J + 1, a fact that cannot be seen directly from the XYZ Lagrangian. The third-lowest scalar primary lies in along multiplet with dimension J + 2−c −3 J −3 + O(J −4), where c −3 is an unknown positive coefficient. The coefficient c −3 is proportional to the leading superconformal interaction term in the effective theory on moduli space. The positivity of c −3 does not follow from supersymmetry, but rather from unitarity of moduli scattering and the absence of superluminal signal propagation in the effective dynamics of the complex modulus. We also prove a general lemma, that scalar semi-short representations form a module over the chiral ring in a natural way, by ordinary multiplication of local operators. Combined with the existence of scalar semi-short states at large J, this proves the existence of scalar semi-short states at all values of J. Thus the combination of $$ \mathcal{N}=2 $$ superconformal symmetry with the large-J expansion is more powerful than the sum of its parts.

Dimension analysis of continuous functions with unbounded variation

Abstract: In this paper, we mainly discuss fractal dimensions of continuous functions with unbounded variation. First, we prove that Hausdorff dimension, Packing dimension and Modified Box-counting dimension of continuous functions containing one UV point are 1. The above conclusion still holds for continuous functions containing finite UV points. More generally, we show the result that Hausdorff dimension of continuous functions containing countable UV points is 1 also. Finally, Box dimension of continuous functions containing countable UV points has been proved to be 1 when f(x) is self-similar.

Scalar curvature on compact complex manifolds

Abstract: In this paper, we prove that, a compact complex manifold $X$ admits a smooth Hermitian metric with positive (resp. negative) scalar curvature if and only if $K_X$ (resp. $K_X^{-1}$) is not pseudo-effective. On the contrary, we also show that on an arbitrary compact complex manifold $X$ with complex dimension $\geq 2$, there exist smooth Hermitian metrics with positive total scalar curvature and one of the key ingredients in the proof relies on a recent solution to the Gauduchon conjecture by G. Szekelyhidi, V. Tosatti and B. Weinkove.

Quasiflats in hierarchically hyperbolic spaces

Abstract: The rank of a hierarchically hyperbolic space is the maximal number of unbounded factors in a standard product region. For hierarchically hyperbolic groups, this coincides with the maximal dimension of a quasiflat. Examples for which the rank coincides with familiar quantities include: the dimension of maximal Dehn twist flats for mapping class groups, the maximal rank of a free abelian subgroup for right-angled Coxeter and Artin groups, and, for the Weil--Petersson metric, the rank is the integer part of half the complex dimension of Teichmuller space. We prove that any quasiflat of dimension equal to the rank lies within finite distance of a union of standard orthants (under a mild condition satisfied by all natural examples). This resolves outstanding conjectures when applied to various examples. For mapping class group, we verify a conjecture of Farb; for Teichmuller space we answer a question of Brock; for CAT(0) cubical groups, we handle special cases including right-angled Coxeter groups. An important ingredient in the proof is that the hull of any finite set in an HHS is quasi-isometric to a CAT(0) cube complex of dimension bounded by the rank. We deduce a number of applications. For instance, we show that any quasi-isometry between HHSs induces a quasi-isometry between certain simpler HHSs. This allows one, for example, to distinguish quasi-isometry classes of right-angled Artin/Coxeter groups. Another application is to quasi-isometric rigidity. Our tools in many cases allow one to reduce the problem of quasi-isometric rigidity for a given hierarchically hyperbolic group to a combinatorial problem. We give a new proof of quasi-isometric rigidity of mapping class groups, which, given our general quasiflats theorem, uses simpler combinatorial arguments than in previous proofs.

Semi-parabolic bifurcations in complex dimension two

Abstract: Parabolic bifurcations in one complex dimension demonstrate a wide variety of interesting dynamical phenomena. In this paper we consider the bifurcations of a holomorphic diffeomorphism in two complex dimensions with a semi-parabolic, semiattracting fixed point.

Operator Dimensions from Moduli

Abstract: We consider the operator spectrum of a three-dimensional ${\cal N} = 2$ superconformal field theory with moduli spaces of one complex dimension, such as the fixed point theory with three chiral superfields $X,Y,Z$ and a superpotential $W = XYZ$. By using the existence of an effective theory on each branch of moduli space, we calculate the anomalous dimensions of certain low-lying operators carrying large $R$-charge $J$. While the lowest primary operator is a BPS scalar primary, the second-lowest scalar primary is in a semi-short representation, with dimension exactly $J+1$, a fact that cannot be seen directly from the $XYZ$ Lagrangian. The third-lowest scalar primary lies in a long multiplet with dimension $J+2 - c_{-3} \, J^{-3} + O(J^{-4})$, where $c_{-3}$ is an unknown positive coefficient. The coefficient $c_{-3}$ is proportional to the leading superconformal interaction term in the effective theory on moduli space. The positivity of $c_{-3}$ does not follow from supersymmetry, but rather from unitarity of moduli scattering and the absence of superluminal signal propagation in the effective dynamics of the complex modulus. We also prove a general lemma, that scalar semi-short representations form a module over the chiral ring in a natural way, by ordinary multiplication of local operators. Combined with the existence of scalar semi-short states at large $J$, this proves the existence of scalar semi-short states at all values of $J$. Thus the combination of ${\cal N}=2$ superconformal symmetry with the large-$J$ expansion is more powerful than the sum of its parts.

Tropical Geometric Compactification of Moduli, II - $A_g$ case and holomorphic limits -

Abstract: We compactify the classical moduli variety $A_g$ of principally polarized abelian varieties of complex dimension $g$ by attaching the moduli of flat tori of real dimensions at most $g$ in an explicit manner. Equivalently, we explicitly determine the Gromov-Hausdorff limits of principally polarized abelian varieties. This work is analogous to the first of our series (available at arXiv:1406.7772v2), which compactified the moduli of curves by attaching the moduli of metrized graphs. Then, we also explicitly specify the Gromov-Hausdorff limits along holomorphic family of abelian varieties and show that they form special non-trivial subsets of the whole boundary. We also do it for algebraic curves case and observe a crucial difference with the case of abelian varieties.

Geometric Arveson-Douglas Conjecture and Holomorphic Extension

Abstract: In this paper we introduce techniques from complex harmonic analysis to prove a weaker version of the Geometric Arveson-Douglas Conjecture for complex analytic subsets that is smooth on the boundary of the unit ball and intersects transversally with it. In fact, we prove that the projection operator onto the corresponding quotient module is in the Toeplitz algebra $\mathcal{T}(L^{\infty})$, which implies the essential normality of the quotient module. Combining some other techniques we actually obtain the $p$-essential normality for $p>2d$, where $d$ is the complex dimension of the analytic subset. Finally, we show that our results apply for the closure of a radical polynomial ideal $I$ whose zero variety satisfies the above conditions. A key technique is defining a right inverse operator of the restriction map from the unit ball to the analytic subset generalizing the result of Beatrous's paper "$L^p$-estimates for extensions of holomorphic functions".

Berezin–toeplitz quantization, hyperkähler manifolds, and multisymplectic manifolds

Abstract: We suggest a way to quantize, using Berezin–Toeplitz quantization, a compact hyperkahler manifold (equipped with a natural 3-plectic form), or a compact integral Kahler manifold of complex dimension n regarded as a (2n−1)-plectic manifold. We show that quantization has reasonable semiclassical properties.

Scalar curvature on compact complex manifolds

Abstract: In this paper, we prove that, a compact complex manifold $X$ admits a smooth Hermitian metric with positive (resp. negative) scalar curvature if and only if $K_X$ (resp. $K_X^{-1}$) is not pseudo-effective. On the contrary, we also show that on an arbitrary compact complex manifold $X$ with complex dimension $\geq 2$, there exist smooth Hermitian metrics with positive total scalar curvature and one of the key ingredients in the proof relies on a recent solution to the Gauduchon conjecture by G. Szekelyhidi, V. Tosatti and B. Weinkove.

Boolean dimension and local dimension

Abstract: Dimension is a standard and well-studied measure of complexity of posets. Recent research has provided many new upper bounds on the dimension for various structurally restricted classes of posets. Bounded dimension gives a succinct representation of the poset, admitting constant response time for queries of the form "is $x

Quaternionic Bott-Chern Cohomology and existence of HKT metrics

Abstract: We study quaternionic Bott-Chern cohomology on compact hypercomplex manifolds and adapt some results from complex geometry to the quaternionic setting. For instance, we prove a criterion for the existence of HKT metrics on compact hypercomplex manifolds of real dimension 8 analogous to the one given by Teleman [35] and Angella-Dloussky-Tomassini [3] for compact complex surfaces.

Polynomial skew-products in dimension 2: Bulging and Wandering Fatou components

Abstract: In this short note we give an updated account of the recent results on Fatou components for polynomial skew-products in complex dimension two in a neighbourhood of an invariant fiber, dividing our discussion according to the different possible kinds of invariant fibers.

The projective ensemble and distribution of points in odd-dimensional spheres

Abstract: We define a determinantal point process on the complex projective space that reduces to the so-called spherical ensemble for complex dimension 1 under identification of the 2-sphere with the Riemann sphere. Through this determinantal point process we propose a point processs in odd-dimensional spheres that produces fairly well-distributed points, in the sense that the expected value of the Riesz 2-energy for these collections of points is smaller than all previously known bounds.

Dimension theory and components of algebraic stacks

Abstract: We prove some basic results on the dimension theory of algebraic stacks, and on the multiplicities of their irreducible components, for which we do not know a reference.

Only distances are required to reconstruct submanifolds

Abstract: In this paper, we give the rst algorithm that outputs a faithful reconstruction of a submanifold of Euclidean space without maintaining or even constructing complicated data structures such as Voronoi diagrams or Delaunay complexes. Our algorithm uses the witness complex and relies on the stability of power protection, a notion introduced in this paper. The complexity of the algorithm depends exponentially on the intrinsic dimension of the manifold, rather than the dimension of ambient space, and linearly on the dimension of the ambient space. Another interesting feature of this work is that no explicit coordinates of the points in the point sample is needed. The algorithm only needs the distance matrix as input, i.e., only distance between points in the point sample as input.

Application of the support function and the steiner point on the study of interference alignment channel

Abstract: Interference is usually viewed as an obstacle to communication in wireless networks. Therefore, we describe a way to find a doubly infinite nested lattice partition chain for the real dimension 2 in order to realize interference alignment onto these lattices and a procedure to find the minimum solution of the mean square error related to the corresponding channel quantization. Besides, we propose a new methodology based on the support function and Steiner points of fuzzy vectors to find the closest point of a coset in a lattice. Such a methodology is flexible enough to combine the fuzzy characterization with the geometric insights and it is able to maintain the inherent stochastic nature of the wireless communication networks.

Cohomologies on Hypercomplex Manifolds

Abstract: We review some cohomological aspects of complex and hypercomplex manifolds and underline the differences between both realms. Furthermore, we try to highlight the similarities between compact complex surfaces on one hand and compact hypercomplex manifolds of real dimension 8 with holonomy of the Obata connection in \(\mathrm{SL}(2, \mathbb{H})\) on the other hand.

Log-canonical thresholds in real and complex dimension 2

Abstract: We study the set of log-canonical thresholds (or critical integrability indices) of holomorphic (resp. real analytic) function germs in $\mathbb{C}^2$ (resp. $\mathbb{R}^2$). In particular, we prove that the ascending chain condition holds, and that the positive accumulation points of decreasing sequences are precisely the integrability indices of holomorphic (resp. real analytic) functions in dimension $1$. This gives a new proof of a theorem of Phong-Sturm.

On the Laplace-Beltrami operator on compact complex spaces

Abstract: Let $(X,h)$ be a compact and irreducible Hermitian complex space of complex dimension $v>1$. In this paper we show that the Friedrichs extension of both the Laplace-Beltrami operator and the Hodge-Kodaira Laplacian acting on functions has discrete spectrum. Moreover we provide some estimates for the growth of the corresponding eigenvalues and we use these estimates to deduce that the associated heat operators are trace-class. Finally we give various applications to the Hodge-Dolbeault operator and to the Hodge-Kodaira Laplacian in the setting of Hermitian complex spaces of complex dimension $2$.

Affine special Kaehler structures in real dimension two

Abstract: We review properties of affine special Kaehler structures focusing on singularities of such structures in the simplest case of real dimension two. We describe all possible isolated singularities and compute the monodromy of the flat symplectic connection, which is a part of a special Kaehler structure, near a singularity. Beside numerous local examples, we construct continuous families of special Kaehler structures with isolated singularities on the projective line.

Complete intersections with s 1 -action

Abstract: We give the diffeomorphism classification of complete intersections with S 1-symmetry in dimension ≤ 6. In particular, we show that a 6-dimensional complete intersection admits a smooth non-trivial S 1-action if and only if it is diffeomorphic to the complex projective space or the quadric. We also prove that in any odd complex dimension only finitely many complete intersections can carry a smooth effective action by a torus of rank > 1.

Algebraic approximation preserving dimension

Abstract: We prove that each semialgebraic subset of \({\mathbb R}^n\) of positive codimension can be locally approximated of any order by means of an algebraic set of the same dimension. As a consequence of previous results, algebraic approximation preserving dimension holds also for semianalytic sets.

A Quasi-isometric Embedding Algorithm

Abstract: The Whitney embedding theorem gives an upper bound on the smallest embedding dimension of a manifold. If a data set lies on a manifold, a random projection into this reduced dimension will retain the manifold structure. Here we present an algorithm to find a projection that distorts the data as little as possible.

Pointed $p^2q$-dimensional Hopf algebras in positive characteristic

Abstract: Let $p,q,r$ be distinct prime numbers and $\mathds{k}$ an algebraically closed field of characteristic $p$. We study the classification of pointed Hopf algebras over $\mathds{k}$ of dimension $p^2q$, $pq^2$ and $pqr$. We obtain a complete classification of pointed Hopf algebras of dimension $pq$, $pqr$, $p^2q$, $2q^2$ and $4p$. We also classify all pointed Hopf algebras of dimension $pq^2$ whose diagrams are Nichols algebras and show that pointed Hopf algebras of dimension $pq^2$ whose diagrams are not Nichols algebras must be Hopf extensions of Taft algebras by restricted universal enveloping algebras of dimension $p$. In particular, we obtain many new non-commutative and non-cocommutative finite-dimensional pointed Hopf algebras.

Teichmüller space of negatively curved metrics on complex hyperbolic manifolds is not contractible

Abstract: We prove that for all n = 4k − 2 and k ≥ 2 there exists a closed smooth complex hyperbolic manifold M with real dimension n having non-trivial π1(T<0(M)). T<0(M) denotes the Teichmuller space of all negatively curved Riemannian metrics on M, which is the topological quotient of the space of all negatively curved metrics modulo the space of self-diffeomorphisms of M that are homotopic to the identity