Space CP Research Articles

Explicit construction of new Moishezon twistor spaces

In this paper we explicitly construct Moishezon twistor spaces on nCP^2 for arbitrary n>1 which admit a holomorphic C*-action. When n=2, they coincide with Y. Poon's twistor spaces. When n=3, they coincide with the one studied by the author in math.DG/0403528. When n>3, they are new twistor spaces, to the best of the author's knowledge. By investigating the anticanonical system, we show that our twistor spaces are bimeromorphic to conic bundles over certain rational surfaces. The latter surfaces can be regarded as orbit spaces of the C*-action on the twistor spaces. Namely they are minitwistor spaces. We explicitly determine their defining equations in CP^4. It turns out that the structure of the minitwistor space is independent of n. Further we concrelely construct a CP^2-bundle over the resolution of this surface, and provide an explicit defining equation of the conic bundles. It shows that the number of irreducible components of the discriminant locus for the conic bundles increases as n does. Thus our twistor spaces have a lot of similarities with the famous LeBrun twistor spaces, where the minitwistor space CP^1 x CP^1 in LeBrun's case is replaced by our minitwistor spaces found in math.DG/0508088.

The conjugate points of ℂℙ∞ and the zeroes of bergman kernel

Two points of the infinite dimensional complex projective space ℂℙ∞ with homogeneous coordinates a = (a0,a1,a2,…) and b = (b0,b1,b2,…), respectively, are conjugate if and only if they are complex orthogonal, i.e., ab†=∑j=0∞ajbj¯=0. For a complete ortho-normal system φ(t)=(φ(t), φ1(t), φ2(t), …) of L2H(D) the space of the holomorphic and absolutely square integrable functions in the bounded domain D of ℂn, φ(t),t ∈ D is considered as the homogeneous coordinate of a point in ℂℙ∞. The correspondence t ↦ φ(t) induces a holomorphic imbedding ιφ:D → ℂℙ∞. It is proved that the Bergman kernel ▪ of D equals to zero for the two points t and v in D if and only if their image points under ιφ are conjugate points of ℂℙ∞.

A WEAKLY ANALYTIC LOCALLY CONVEX SPACE WHICH IS NOT K-ANALYTIC

AbstractIt is shown that the dual of the space Cp(I) of all real-valued continuous functions on the closed unit interval with the pointwise topology, when equipped with the Mackey topology, is a non K-analytic but weakly analytic locally convex space.

On Fubini–Study form and Reidemeister torsion

For a general chain complex ( C * , ∂ * ) , one can associate the Reidemeister torsion of it. We prove the relation between Reidemeister torsion and Fubini–Study 2-form ω FS of the complex projective space CP n .

Quantisation of Twistor Theory by Cocycle Twist

We present the main ingredients of twistor theory leading up to and including the Penrose-Ward transform in a coordinate algebra form which we can then `quantise' by means of a functorial cocycle twist. The quantum algebras for the conformal group, twistor space CP^3, compactified Minkowski space CMh and the twistor correspondence space are obtained along with their canonical quantum differential calculi, both in a local form and in a global *-algebra formulation which even in the classical commutative case provides a useful alternative to the formulation in terms of projective varieties. We outline how the Penrose-Ward transform then quantises. As an example, we show that the pull-back of the tautological bundle on CMh pulls back to the basic instanton on S^4\subset CMh and that this observation quantises to obtain the Connes-Landi instanton on \theta-deformed S^4 as the pull-back of the tautological bundle on our \theta-deformed CMh. We likewise quantise the fibration CP^3--> S^4 and use it to construct the bundle on \theta-deformed CP^3 that maps over under the transform to the \theta-deformed instanton.

A NOTE ON SPACES Cp(X)K-ANALYTIC-FRAMED IN ℝX

AbstractThis paper characterizes the K-analyticity-framedness in ℝX for Cp(X) (the space of real-valued continuous functions on X with pointwise topology) in terms of Cp(X). This is used to extend Tkachuk’s result about the K-analyticity of spaces Cp(X) and to supplement the Arkhangel ′skiĭ–Calbrix characterization of σ-compact cosmic spaces. A partial answer to an Arkhangel ′skiĭ–Calbrix problem is also provided.

One more variation of the point-open game

A topological game Dense G��-sets (also denoted by DG) is introduced as follows: for any n 2 ! at the n-th move the player I takes a point xn 2 X and II responds by taking a G�-set Qn in the space X such that xn 2 Qn. The play stops after ! moves and I wins if the set S {Qn : n 2 !} is dense in X. Otherwise the player II is declared to be the winner. We study classes of spaces on which the player I has a winning strategy. It is evident that the I-favorable spaces constitute a generalization of the class of separable spaces. We show that there exists a neutral space for the game DG and prove, among other things, that Lindelof scattered spaces and dyadic spaces are I-favorable. We characterize I- favorability for the game DG in the spaces Cp(X); one of the applications is that, for a Lindelof �-space X, the space Cp(X) is I-favorable for DG if and only if X is !-monolithic.

A second look at 𝒩 = 1 supersymmetric AdS4vacua of type IIA supergravity

We show that a class of type IIA vacua recently found within the D=4 effec- tive approach corresponds to compactification on AdS 4 ×S 3 ×S 3 /Z 3 . The results obtained using the effective method completely match the general ten-dimensional analysis for the existence of N=1 warped compactifications on AdS 4 ×M6. In particular, we verify that the internal metric is nearly-Kahler and that for specific values of the parameters the Bianchi identity of the RR 2-form is fulfilled without sources. For another range of parameters, in- cluding the massless case, the Bianchi identity is satisfied when D6-branes are introduced. Solving the tadpole cancellation conditions in D=4 we are able to find examples of appro- priate sets of branes. In the second part of this paper we describe how an example with internal space CP 3 but with non nearly-Kahler metric fits into the general analysis of flux vacua.

Lindelöf spaces C(X) over topological groups

Theorem 1 proves (among the others) that for a locally compact topological group X the following assertions are equivalent: (i) X is metrizable and σ-compact. (ii) Cp(X) is analytic. (iii) Cp(X) is K-analytic. (iv) Cp(X) is Lindelöf. (v) Cc(X) is a separable metrizable and complete locally convex space. (vi) Cc(X) is compactly dominated by irrationals. This result supplements earlier results of Corson, Christensen and Calbrix and provides several applications, for example, it easily applies to show that: (1) For a compact topological group X the Eberlein, Talagrand, Gul'ko and Corson compactness are equivalent and any compact group of this type is metrizable. (2) For a locally compact topological group X the space Cp(X) is Lindelöf iff Cc(X) is weakly Lindelöf. The proofs heavily depend on the following result of independent interest: A locally compact topological group X is metrizable iff every compact subgroup of X has countable tightness (Theorem 2). More applications of Theorem 1 and Theorem 2 are provided.

Riemannian flag manifolds with homogeneous geodesics

A geodesic in a Riemannian homogeneous manifold (M = G/K, g) is called a homogeneous geodesic if it is an orbit of a one-parameter subgroup of the Lie group G. We investigate G-invariant metrics with homogeneous geodesics (i.e., such that all geodesics are homogeneous) when M = G/K is a flag manifold, that is, an adjoint orbit of a compact semisimple Lie group G. We use an important invariant of a flag manifold M = G/K, its T-root system, to give a simple necessary condition that M admits a non-standard G-invariant metric with homogeneous geodesics. Hence, the problem reduces substantially to the study of a short list of prospective flag manifolds. A common feature of these spaces is that their isotropy representation has two irreducible components. We prove that among all flag manifolds M = G/K of a simple Lie group G, only the manifold Com(R 2l+2 ) = SO(2l +1)/U(l) of complex structures in R 2l+2 , and the complex projective space CP 2l-1 = Sp(l)/U(1) Sp(l- 1) admit a non-naturally reductive invariant metric with homogeneous geodesics. In all other cases the only G-invariant metric with homogeneous geodesics is the metric which is homothetic to the standard metric (i.e., the metric associated to the negative of the Killing form of the Lie algebra g of G). According to F. Podesta and G.Thorbergsson (2003), these manifolds are the only non-Hermitian symmetric flag manifolds with coisotropic action of the stabilizer.

Kähler geometry of hyperbolic type on the manifold of nondegenerate m-pairs

A nondegenerate m-pair (A, Ξ) in an n-dimensional projective space ℝP n consists of an m-plane A and an (n − m − 1)-plane Ξ in ℝP n , which do not intersect. The set \(\mathfrak{N}_m^n \) of all nondegenerate m-pairs ℝP n is a 2(n − m)(n − m − 1)-dimensional, real-complex manifold. The manifold \(\mathfrak{N}_m^n \) is the homogeneous space \(\mathfrak{N}_m^n = {{GL(n + 1,\mathbb{R})} \mathord{\left/ {\vphantom {{GL(n + 1,\mathbb{R})} {GL(m + 1,\mathbb{R})}}} \right. \kern-\nulldelimiterspace} {GL(m + 1,\mathbb{R})}} \times GL(n - m,\mathbb{R})\) equipped with an internal Kähler structure of hyperbolic type. Therefore, the manifold \(\mathfrak{N}_m^n \) is a hyperbolic analogue of the complex Grassmanian ℂG m,n = U(n+1)/U(m+1) × U(n−m). In particular, the manifold of 0-pairs \(\mathfrak{N}_m^n {{GL(n + 1,\mathbb{R})} \mathord{\left/ {\vphantom {{GL(n + 1,\mathbb{R})} {GL(1,\mathbb{R})}}} \right. \kern-\nulldelimiterspace} {GL(1,\mathbb{R})}} \times GL(n,\mathbb{R})\) is a hyperbolic analogue of the complex projective space ℂP n = U(n+1)/U(1) × U(n). Similarly to ℂP n , the manifold \(\mathfrak{N}_m^n \) is a Kähler manifold of constant nonzero holomorphic sectional curvature (relative to a hyperbolic metrics). In this sense, \(\mathfrak{N}_0^n \) is a hyperbolic spatial form. It was proved in [6] that the manifold of 0-pairs \(\mathfrak{N}_0^n \) is globally symplectomorphic to the total space T*ℝP n of the cotangent bundle over the projective space ℝP n . A generalization of this result (see [7]) is as follows: the manifold of nondegenerate m-pairs \(\mathfrak{N}_m^n \) is globally symplectomorphic to the total space T*ℝG m,n of the cotangent bundle over the Grassman manifold ℝG m,n of m-dimensional subspaces of the space ℝP n .In this paper, we study the canonical Kähler structure on \(\mathfrak{N}_m^n \). We describe two types of submanifolds in \(\mathfrak{N}_m^n \), which are natural hyperbolic spatial forms holomorphically isometric to manifolds of 0-pairs in ℝP m +1 and in ℝP n−m , respectively. We prove that for any point of the manifold \(\mathfrak{N}_m^n \), there exist a 2(n − m)-parameter family of 2(m + 1)-dimensional hyperbolic spatial forms of first type and a 2(m + 1)-parameter family of 2(n − m)-dimensional hyperbolic spatial forms of second type passing through this point. We also prove that natural hyperbolic spatial forms of first type on \(\mathfrak{N}_m^n \) are in bijective correspondence with points of the manifold \(\mathfrak{N}_{m + 1}^n \) and natural hyperbolic spatial forms of second type on \(\mathfrak{N}_m^n \) are in bijective correspondence with points of the manifolds \(\mathfrak{N}_{m + 1}^n \).

A selective version of the property of Reznichenko in function spaces

We consider the selective version of the property of Reznichenko and find necessary and sufficient conditions under which the function space Cp(X) over a Tychonoff space X satisfies this property. The dual description involves selection principles and game theory.

The Dual Complexity Space as the Dual of a Normed Cone

In [M. Schellekens, The Smyth completion: A common foundation for denotational semantics and complexity analysis, in: Proc. MFPS 11, Electronic Notes in Theoretical Computer Science, vol. 1 (1995), 23 pages] M. Schellekens introduced the complexity (quasi-metric) space as a part of the research in Theoretical Computer Science and Topology, with applications to the complexity analysis of algorithms. Later on, S. Romaguera and M. Schellekens ([S. Romaguera, M. Schellekens, Quasi-metric properties of complexity spaces, Topology Appl. 98 (1999), 311–322]) introduced the so-called dual complexity (quasi-metric) space and established several quasi-metric properties of the complexity space via the analysis of th e dual. These authors also proved in [S. Romaguera, M. Schellekens, Duality and quasi-normability for complexity spaces, Appl. Gen. Topology 3 (2002), 91–112] that actually the dual complexity space C∗ can be modeled as a norm-weightable cone whose induced quasi-metric is Smyth complete. This fact suggests the existence of deep connections between a general theory of (dual) complexity spaces and Asymmetric Functional Analysis. These connections have been recently explored in [L.M. Garcá-Raffi, S. Romaguera, E.A. Sánchez-Pérez, Sequence spaces and asymmetric norms in the theory of compuational complexity, Math. Comput. Model 36 (2002), 1–11], [L.M. Garcá-Raffi, S. Romaguera, E.A. Sánchez-Pérez, The supremum asymmetric norm on sequence spaces: a general framework to measure complexity distances, in: Proceedings of the Second Irish Conference on the Mathematical Foundations of Computer Science and Information Technology (MFCSIT 2002), Galway, Ireland, July 2002; Electronic Notes in Theoret. Comput. Sci. 74 (2003), URL: http://www.elsevier.nl/locate/entcs/volume74.htm 12 pages] and [M. O'Keefe, S. Romaguera, M. Schellekens, Norm-weightable Riesz spaces and the dual complexity space, in: Proceedings of the Second Irish Conference on the Math ematical Foundations of Computer Science and Information Technology (MFCSIT 2002), Galway, Ireland, July 2002; Electronic Notes in Theoret. Comput. Sci. 74 (2003), URL: http://www.elsevier.nl/locate/entcs/volume74.htm 17 pages]. In particular, it was proved in [L.M. Garcá-Raffi, S. Romaguera, E.A. Sánchez-Pérez, Sequence spaces and asymmetric norms in the theory of compuational complexity, Math. Comput. Model 36 (2002), 1–11] that the so-called dual p-complexity space Cp∗, with 1⩽p<∞, is isometrically isomorphic to the positive cone of the classical Banach space lp. The space C1∗ is exactly the dual complexity space, and thus it is isometrically isomorphic to the positive cone of the Banach space l1 of all absolutely summable real sequences. Here, we continue the analysis of the structure of the dual complexity space C∗. We show that it is the dual space of the positive c one of the Banach space c0 of all real sequences converging to zero, and that its dual space is the positive cone of the Banach space l∞ of all bounded real sequences. Furthermore, the dual space of Cp∗, 1<p<∞, is Cq∗ where 1/p+1/q=1. These results extend to this setting well-known theorems of the classical theory of Functional Analysis.

Some Properties of the Functor O β

It is proved that the space Oβ(X) of weakly additive order-preserving normed functionals with compact supports is a convex subset of the space Cp(Cb(X)). Bibliography: 4 titles.

Formula omitted]-modules on the complex projective space [formula omitted] associated to a quadric

We give a combinatorial description of regular holonomic systems on the complex projective space CP n − 1 with characteristic variety the union of the zero section and the conormal bundle of a smooth quadric (equivalently: those that admit an infinitesimal action of PO ( n ) ). To cite this article: P. Nang, C. R. Acad. Sci. Paris, Ser. I 342 (2006).

Spaces of continuous step functions over LOTS

We investigate spaces Cp(·,n) over LOTS (linearly ordered topological spaces). We find natural necessary conditions for linear Lindelofness of Cp(·,n) over LOTS. We also characterize countably compact LOTS whose Cp(·,n) is linearly Lindelof for each n. Both the necessary conditions and the characterization are given in terms of the topology of the Dedekind completion of a LOTS.

A splitting result for the free loop space of spheres and projective spaces

Let X be a 1-connected compact space such that the algebra H ∗ (X; F2) is generated by one single element. We compute the cohomology of the free loop space H ∗ (�X; F2) including the Steenrod algebra action. When X is a projective space CP n , HP n , the Cayley projective plane CaP 2 or a sphere S m we obtain a splitting result for integral and mod two cohomology of the suspension spectrum � ∞ (�X)+. The splitting is in terms of � ∞ X+ and the Thom spaces Th(q�), q ≥ 0 of the q-fold Whitney sums of the tangent bundle � over X.

A description of (Cp[Lp(M)], Rp[Lp(M)])θ

We give a simple explicit description of the norm in the complex interpolation space (Cp[Lp(M)], Rp[Lp(M)])θ for any von Neumann algebra M and any 1 ≤ p ≤ ∞.

Quantum Hall effect on higher-dimensional spaces

We analysis the quantum Hall effect exhibited by a system of particles moving in a higher-dimensional space. This can be done by considering particles on the Bergman ball B ρ d of radius ρ in the presence of an external magnetic field B and investigate its basic features. Solving the corresponding Hamiltonian to get the energy levels as well as the eigenfunctions. This can be used to study quantum Hall effect of confined particles in the lowest Landau level where density of particles and two point functions are calculated. We take advantage of the symmetry group of the Hamiltonian on B ρ d to make link to the Landau problem analysis on the complex projective spaces CP d . In the limit ρ → ∞ , our analysis coincides with that corresponding to particles on the flat geometry C d . This task has been done for d = 1 , 2 and finally for the generic case, i.e. d ⩾ 3 .

A space Cp(X) is dominated by irrationals if and only if it is K-analytic

We prove that, for any Tychonoff X, the space C p (X) is K-analytic if and only if it has a compact cover {K p : p ∈ ωω} such that K p subset K q whenever p,q ∈ ωω and p ≤ q. Applying this result we show that if C p (X) is K-analytic then C p (υX) is K-analytic as well. We also establish that a space C p (X) is K-analytic and Baire if and only if X is countable and discrete.