Showing papers on "Complex dimension published in 2011"

Fractal Geometry and Number Theory: Complex Dimensions of Fractal Strings and Zeros of Zeta Functions

Abstract: Number theory and fractal geometry are combined in this study of the vibrations of fractal strings. The book centres around a notion of complex dimension, originally developed for the proof of the Prime Number Theorem, and extended here to apply to the zeta functions associated with fractals.

Dimensionality reduction: beyond the Johnson-Lindenstrauss bound

Abstract: Dimension reduction of metric data has become a useful technique with numerous applications. The celebrated Johnson-Lindenstrauss lemma states that any n-point subset of Euclidean space can be embedded in O(e−2 log n)-dimension with (1 + e)-distortion. This bound is known to be nearly tight.In many applications the demand that all distances should be nearly preserved is too strong. In this paper we show that indeed under natural relaxations of the goal of the embedding, an improved dimension reduction is possible where the target dimension is independent of n. Our main result can be viewed as a local dimension reduction. There are a variety of empirical situations in which small distances are meaningful and reliable, but larger ones are not. Such situations arise in source coding, image processing, computational biology, and other applications, and are the motivation for widely-used heuristics such as Isomap and Locally Linear Embedding.Pursuing a line of work begun by Whitney, Nash showed that every C1 manifold of dimension d can be embedded in R2d+2 in such a manner that the local structure at each point is preserved isometrically. Our work is an analog of Nash's for discrete subsets of Euclidean space. For perfect preservation of infinitesimal neighborhoods we substitute near-isometric embedding of neighborhoods of bounded cardinality.We show that any finite subset of Euclidean space can be embedded in O(e−2 log k)-dimension while preserving with (1 + e)-distortion the distances within a "core neighborhood" of each point. (The core neighborhood is a metric ball around the point, whose radius is a substantial fraction of the radius of the ball of cardinality k, the k-neighborhood.) When the metric space satisfies a weak growth rate property, the guarantee applies to the entire k-neighborhood (with some dependency of the embedding dimension on the growth rate). We also show how to obtain a global embedding that also keeps distant points well-separated (at the cost of dependency on the doubling dimension of the space).As an application of our methods we obtain an (Assouad-style) dimension reduction for finite subsets of Euclidean space where the metric is raised to some fractional power (the resulting metrics are known as snowflakes). We show that any such metric X can be embedded in dimension O(e−3 dim(X)) with 1 + e distortion, where dim(X) is the doubling dimension, a measure of the intrinsic dimension of the set. This result improves recent work by Gottlieb and Krauthgamer [20] to a nearly tight bound.The new dimension reduction results are useful for applications such as clustering and distance labeling.

Trihyperkahler reduction and instanton bundles on CP^3

Abstract: A trisymplectic structure on a complex 2n-manifold is a triple of holomorphic symplectic forms such that any linear combination of these forms has constant rank 2n, n or 0, and degenerate forms in $\Omega$ belong to a non-degenerate quadric hypersurface. We show that a trisymplectic manifold is equipped with a holomorphic 3-web and the Chern connection of this 3-web is holomorphic, torsion-free, and preserves the three symplectic forms. We construct a trisymplectic structure on the moduli of regular rational curves in the twistor space of a hyperkaehler manifold, and define a trisymplectic reduction of a trisymplectic manifold, which is a complexified form of a hyperkaehler reduction. We prove that the trisymplectic reduction in the space of regular rational curves on the twistor space of a hyperkaehler manifold M is compatible with the hyperkaehler reduction on M. As an application of these geometric ideas, we consider the ADHM construction of instantons and show that the moduli space of rank r, charge c framed instanton bundles on CP^3 is a smooth, connected, trisymplectic manifold of complex dimension 4rc. In particular, it follows that the moduli space of rank 2, charge c instanton bundles on CP^3 is a smooth complex manifold dimension 8c-3, thus settling a 30-year old conjecture.

Fano manifolds of Calabi-Yau type

Abstract: We introduce and we study a class of odd dimensional compact complex manifolds whose Hodge structure in middle dimension looks like that of a Calabi-Yau threefold. We construct several series of interesting examples from rational homogeneous spaces with special properties.

Contact pairs and locally conformally symplectic structures

Abstract: We discuss a correspondence between certain contact pairs on the one hand, and certain locally conformally symplectic forms on the other. In particular, we characterize these structures through suspensions of contactomorphisms. If the contact pair is endowed with a normal metric, then the corresponding lcs form is locally conformally Kaehler, and, in fact, Vaisman. This leads to classification results for normal metric contact pairs. In complex dimension two we obtain a new proof of Belgun's classification of Vaisman manifolds under the additional assumption that the Kodaira dimension is non-negative. We also produce many examples of manifolds admitting locally conformally symplectic structures but no locally conformally Kaehler ones.

A Note on Gorenstein Flat Dimension

Abstract: Unlike the Gorenstein projective and injective dimensions, the majority of results on the Gorenstein flat dimension have been established only over Noetherian (or coherent) rings. Naturally, one would like to generalize these results to any associative ring. In this direction, we show that the Gorenstein flat dimension is a refinement of the classical flat dimension over any ring; and we investigate the relations between the Gorenstein projective dimension and the Gorenstein flat dimension.

On Lagrangian---Grassmannian codes

Abstract: Using the Lagrangian---Grassmannian, a smooth algebraic variety of dimension n(n + 1)/2 that parametrizes isotropic subspaces of dimension n in a symplectic vector space of dimension 2n, we construct a new class of linear codes generated by this variety, the Lagrangian---Grassmannian codes. We explicitly compute their word length, give a formula for their dimension and an upper bound for the minimum distance in terms of the dimension of the Lagrangian---Grassmannian variety.

The topology of toric symplectic manifolds

Abstract: This is a collection of results on the topology of toric symplectic manifolds. Using an idea of Borisov, we show that a closed symplectic manifold supports at most a finite number of toric structures. Further, the product of two projective spaces of complex dimension at least two (and with a standard product symplectic form) has a unique toric structure. We then discuss various constructions, using wedging to build a monotone toric symplectic manifold whose center is not the unique point displaceable by probes, and bundles and blow ups to form manifolds with more than one toric structure. The bundle construction uses the McDuff–Tolman concept of mass linear function. Using Timorin’s description of the cohomology algebra via the volume function we develop a cohomological criterion for a function to be mass linear, and explain its relation to Shelukhin’s higher codimension barycenters.

Twistor geometry and warped product orthogonal complex structures

Abstract: The twistor space of the sphere S2n is an isotropic Grassmannian that fibers over S2n. An orthogonal complex structure (OCS) on a subdomain of S2n (a complex structure compatible with the round metric) determines a section of this fibration with holomorphic image. In this article, we use this correspondence to prove that any finite energy OCS on R6⊂S6 must be of a special warped product form, and we also prove that any OCS on R2n that is asymptotically constant must itself be constant. We give examples defined on R2n which have infinite energy and examples of nonstandard OCSs on flat tori in complex dimension 3 and greater.

Bounding scalar curvature for global solutions of the Kahler-Ricci flow

Abstract: We show that the scalar curvature is uniformly bounded for the normalized Kahler-Ricci flow on a Kahler manifold with semi-ample canonical bundle. In particular, the normalized Kahler-Ricci flow has long time existence if and only if the scalar curvature is uniformly bounded, for Kahler surfaces, projective manifolds of complex dimension three, and for projective manifolds of all dimensions if assuming the abundance conjecture.

On the holomorphic closure dimension of real analytic sets

Abstract: Given a real analytic (or, more generally, semianalytic) set R in Cn (viewed as R2n), there is, for every p ∈ R, a unique smallest complex analytic germ Xp that contains the germ Rp. We call dimC Xp the holomorphic closure dimension of R at p. We show that the holomorphic closure dimension of an irreducible R is constant on the complement of a closed proper analytic subset of R, and we discuss the relationship between this dimension and the CR dimension of R.

On Quantization of Complex Symplectic Manifolds

Abstract: Let X be a complex symplectic manifold. By showing that any Lagrangian subvariety has a unique lift to a contactification, we associate to X a triangulated category of regular holonomic microdifferential modules. If X is compact, this is a Calabi-Yau category of complex dimension dim X + 1. We further show that regular holonomic microdifferential modules can be realized as modules over a quantization algebroid canonically associated to X.

Constructive dimension and Hausdorff dimension: the case of exact dimension

Abstract: The present paper generalises results by Lutz and Ryabko. We prove a martingale characterisation of exact Hausdorff dimension. On this base we introduce the notion of exact constructive dimension of (sets of) infinite strings. Furthermore, we generalise Ryabko's result on the Hausdorff dimension of the set of strings having asymptotic Kolmogorov complexity = a to the case of exact dimension.

One-dimensional lie foliations with generic singularities in complex dimension three

Abstract: Abstract We prove that a germ of a one-dimensional holomorphic foliation with a generic singularity in dimension two or three that exhibits a Lie group transverse structure in the complement of some codimension one analytic subset is logarithmic, that is, given by a system of closed meromorphic one-forms with simple poles. In the global context, we prove that a foliation by curves in a three-dimensional complex manifold with generic singularities and a Lie group transverse structure off a codimension one analytic subset is logarithmic; that is, it is given by a system of closed meromorphic one-forms with simple poles.

Complete Intersections with S^1-action

Abstract: We give the diffeomorphism classification of complete intersections with S^1-symmetry in dimension less than or equal to 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 $>1$.

Higher Secants of Spinor Varieties

Abstract: Let $S_h$ be the even pure spinors variety of a complex vector space $V$ of even dimension $2h$ endowed with a non degenerate quadratic form $Q$ and let $\sigma_k(S_h) $ be the $k$-secant variety of $S_h$. We decribe a probabilistic algorithm which computes the complex dimension of $\sigma_k(S_h) $. Then, by using an inductive argument, we get our main result: $\sigma_3(S_h) $ has the expected dimension except when $h\in \{7,8\} $. Also we provide theoretical arguments which prove that $S_7$ has a defective 3-secant variety and $S_8$ has defective 3-secant and 4-secant varieties.

Complex Plateau problem: old and new results and prospects

Abstract: The complex Plateau problem is analogous, in a Hermitian complex manifold, to the classical Plateau problem in 3 dimensional real space: it is a geometrical problem of extension of a closed real manifold into a complex analytic subvariety, or into a Levi-flat subvariety. Minimality of complex analytic subvarieties and analogous properties of Levi-flat subvarieties, in K\"ahler manifolds, are recalled or given. Known results in n dimensional complex and projective complex spaces are recalled. Extensions to real parametric problems are solved or proposed, leading to the construction of Levi-flat hypersurfaces with prescribed boundary in some complex manifolds.

Calabi Energies of Extremal Toric Surfaces

Abstract: We derive a formula for the L^2 norm of the scalar curvature of any extremal Kaehler metric on a compact toric manifold, stated purely in terms of the geometry of the corresponding moment polytope. The main interest of this formula pertains to the case of complex dimension 2, where it plays a key role in construction of Bach-flat metrics on appropriate 4-manifolds.

Extremal length boundary of Teichm\"uller space contains non-Busemann points

Abstract: In this paper, we shall show that the metric boundary of the Teichmueller space with respect to the Teichmueller distance contains non-Busemann points when the complex dimension of the Teichmueller space is at least two.

Bergman kernel on toric Kahler manifolds

Abstract: Let $(L,h)\to (X, \omega)$ be a compact toric polarized Kahler manifold of complex dimension $n$. For each $k\in N$, the fibre-wise Hermitian metric $h^k$ on $L^k$ induces a natural inner product o ...

On the asymptotic extension dimension

Abstract: We introduce an asymptotic counterpart of the extension dimension defined by Dranishnikov. The main result establishes the relationship between the asymptotic extensional dimension of a proper metric space and the extension dimension of its Higson corona.

Invariant stably complex structures on topological toric manifolds

Abstract: We show that any $(\C ^*)^n$-invariant stably complex structure on a topological toric manifold of dimension $2n$ is integrable. We also show that such a manifold is weakly $(\C ^*)^n$-equivariantly isomorphic to a toric manifold.

Random Riesz energies on compact K\"{a}hler manifolds

Abstract: This article determines the asymptotics of the expected Riesz s-energy of the zero set of a Gaussian random systems of polynomials of degree N as the degree N tends to infinity in all dimensions and codimensions. The asymptotics are proved more generally for sections of any positive line bundle over any compact Kaehler manifold. In comparison with the results on energies of zero sets in one complex dimension due to Qi Zhong (arXiv:0705.2000) (see also [arXiv:0705.2000]), the zero sets have higher energies than randomly chosen points in dimensions > 2 due to clumping of zeros.

Topological Methods in Stein Geometry

Abstract: In this chapter we introduce more advanced topological methods to the study of geometric problems on Stein manifolds and to Oka theory. We begin by considering complex points of smooth real surfaces in complex surfaces. After proving the Lai formulae and the Eliashberg-Harlamov cancellation theorem, we explore connections between the generalized adjunction inequality and the existence of embedded or immersed surfaces with a Stein neighborhood basis in a given complex surface. We then show how the Seiberg-Witten theory bears upon these questions by arguments similar to those leading to the proof of the generalized Thom conjecture. In the second part we present the Eliashberg-Gompf construction of integrable Stein structures on almost complex manifolds with a suitable handlebody decomposition, and we prove the soft Oka principle to the effect that every continuous map from a Stein manifold \(X\) to an arbitrary complex manifold \(Y\) is homotopic to a holomorphic map provided that we allow homotopic deformations of the Stein structure on \(X\) and, in real dimension four, also a change of the underlying smooth structure on \(X\).

A Invertible Dimension Reduction of Curves on a Manifold

Abstract: In this paper, we propose a novel lower dimensional representation of a shape sequence. The proposed dimension reduction is invertible and computationally more efficient in comparison to other related works. Theoretically, the differential geometry tools such as moving frame and parallel transportation are successfully adapted into the dimension reduction problem of high dimensional curves. Intuitively, instead of searching for a global flat subspace for curve embedding, we deployed a sequence of local flat subspaces adaptive to the geometry of both of the curve and the manifold it lies on. In practice, the experimental results of the dimension reduction and reconstruction algorithms well illustrate the advantages of the proposed theoretical innovation.