Dimensional Complex Projective Space Research Articles

The scalar curvature of CR submanifold of maximal CR dimension with a conformal change

In this paper, we investigate the codimension reduction theorem for an n-dimensional submanifold of an (n + p)-dimensional manifold (M, ~g) with a conformal change g = exp(–f)g where g denotes the Fubini-study metric on (n + p)-dimensional complex projective space P n+p/2 (C). Moreover, we tend to calculate the scalar curvature of an n-dimensional CR submanifold of maximal CR dimension of (M; g) and achieve the sufficient conditions for the existence of a totally geodesic submanifold M0 that includes M.

A Generalization of Lappan’s Theorem to Higher Dimensional Complex Projective Space

A Generalization of Lappan’s Theorem to Higher Dimensional Complex Projective Space

RUANG PROYEKTIF KOMPLEKS 〖CP〗^n ADALAH MANIFOLD KOMPLEKS

<p>The purpose of this paper to determine the complex projective space as a complex manifold is to calculate the cohomology of the coherent sheaves of . The research method in this paper is to construct an -dimensional complex projective space, namely and then the n-dimensional complex projective space, namely , is a complex manifold. The result of this research is the -dimensional complex projective space, namely is a complex and compact manifold.</p>

REAL K-COHOMOLOGY OF COMPLEX PROJECTIVE SPACES

In this paper, we determine the structure of KO-cohomology of complex projective space P l and its product space P l × P m as algebras over the coefficient ring KO ∗ . We also give a description of the map KO ∗ ( P l+m ) → KO ∗ ( P l × P m ) induced by the map that classifies the tensor product of the canonical line bundles and show that its image is not contained in the image of the cross product KO ∗ ( P l ) ⊗KO∗ KO ∗ ( P m ) → KO ∗ ( P l × P m ) to see that non-existence of the formal group structure on KO ∗ ( P ∞ ). Introduction A commutative ring spectrum E is said to be complex oriented if an element x of the reduced E-cohomology of the infinite dimensional complex projective space CP ∞ is given such that x maps to a generator of the reduced E-cohomology of 1-dimensional complex projective space CP 1 ((2)). We call such an element x a complex orientation of E. On the other hand, if E-homology E∗E of E is a flat over the coefficient ring E∗, E∗E has a structure of a Hopf algebroid and E-homology theory takes values in the category of E∗E-comodule, in other words, the category of representations of the groupoid represented by the affine groupoid scheme represented by E∗E ((1)). If E is a complex oriented ring spectrum, the E-cohomology of the complex projective space is just a truncated polynomial algebra over E∗ and it is shown that E-homology E∗E of E is a flat over E∗. Moreover the product structure of CP ∞ gives a one dimensional formal group law over E ∗ ((5)) which closely relates with the structure of the Hopf algebroid ((2)). The complex K-theory is one of the most basic examples of complex oriented cohomology theories. However, KO-spectrum representing the real K-theory is one of a few well-known examples of spectra E without any complex orientation such that E∗E is flat over E∗ ((2), (7)). In fact, we see that KO-spectrum does not have any complex orientation by showing that the Atiyah-Hirzebruch spectral sequence converging to KO ∗ (CP l ) has a non-trivial

Non-Lorentzian M5-brane theories from holography

M-theory on AdS7 × S4 admits a description where the AdS7 factor is constructed as a timelike Hopf fibration over a non-compact three dimensional complex projective space {tilde{mathrm{mathbb{CP}}}}^3 [1]. We consider the worldvolume theory for M5-branes at a fixed {tilde{mathrm{mathbb{CP}}}}^3 radius which, after reduction along the timelike fibre, is given by an Ω-deformed Yang-Mills theory with eight supercharges. Taking the radius to infinity then induces a classical RG flow. We construct the fixed point action which has an enhanced 24 supercharges and which can be understood as the (2, 0) theory of M5-branes on flat space reduced along a compact null Killing direction.

Characterizations of three homogeneous real hypersurfaces in a complex projective space

In an $n$-dimensional complex hyperbolic space $\mathbb{C}H^n(c)$ of constant holomorphic sectional curvature $c (\lt 0)$, the horosphere HS, which is defined by ${\rm HS} = \lim_{r\to\infty}G(r)$, is one of nice examples in the class of real hypersurfaces. Here, $G(r)$ is a geodesic sphere of radius $r$ $(0 \lt r \lt \infty)$ in $\mathbb{C}H^n(c)$. The second author ([14]) gave a geometric characterization of HS. In this paper, motivated by this result, we study real hypersurfaces $M^{2n-1}$ isometrically immersed into an $n$-dimensional complex projective space $\mathbb{C}P^n(c)$ of constant holomorphic sectional curvature $c(\gt 0)$.

Quantization and explicit diagonalization of new compactified trigonometric Ruijsenaars–Schneider systems

Recently, Feh\'er and Kluck discovered, at the level of classical mechanics, new compactified trigonometric Ruijsenaars-Schneider $n$-particle systems, with phase space symplectomorphic to the $(n-1)$-dimensional complex projective space. In this article, we quantize the so-called type (i) instances of these systems and explicitly solve the joint eigenvalue problem for the corresponding quantum Hamiltonians by generalising previous results of van Diejen and Vinet. Specifically, the quantum Hamiltonians are realized as discrete difference operators acting in a finite-dimensional Hilbert space of complex-valued functions supported on a uniform lattice over the classical configuration space, and their joint eigenfunctions are constructed in terms of discretized $A_{n-1}$ Macdonald polynomials with unitary parameters.

Projectively induced rotation invariant Kähler metrics

We classify Kahler–Einstein manifolds admitting a Kahler immersion into a finite dimensional complex projective space endowed with the Fubini–Study metric, whose codimension is less than or equal to 3 and whose metric is rotation invariant.

Strongly not relatives Kähler manifolds

Abstract In this paper we study Kähler manifolds that are strongly not relative to any projective Kähler manifold, i.e. those Kähler manifolds that do not share a Kähler submanifold with any projective Kähler manifold even when their metric is rescaled by the multiplication by a positive constant. We prove two results which highlight some relations between this property and the existence of a full Kähler immersion into the infinite dimensional complex projective space. As application we get that the 1-parameter families of Bergman-Hartogs and Fock-Bargmann-Hartogs domains are strongly not relative to projective Kähler manifolds.

A Geometrical Perspective on the Coherent Multimode Optical Field and Mode Coupling Equations

The generalization of the Poincare sphere to $N \ge 2$ modes is the ( $N-1$ )-dimensional complex projective space CP( $N-1$ ). There is a minimal set of $2N-2$ Stokes vector components that determine the coherent multimode optical field. These are obtained from the inverse stereographic projection of coordinate hyperplanes in CP( $N-1$ ) into a $2N-2$ sphere, just as in the $N=2$ case. We derive $N$ -mode analogs of Poole’s optical fiber polarization-mode dispersion (PMD) equations that involve only $2N-2$ independent variables. This is achieved by means of an explicit generalized coherent state representation of the optical field, which enables the components of the PMD vector to be expressed in terms of the optical state and its frequency derivatives. Poole’s equations describe mode coupling as a flow on CP( $N-1$ ). We give general constraints on the mode-coupling matrix and Stokes vector components. The group delay operator is shown to be a rank-2 perturbation of a diagonal matrix.

On holomorphic isometric immersions of nonhomogeneous Kähler–Einstein manifolds into the infinite dimensional complex projective space

In the paper, we study the existence of holomorphic isometric immersions from nonhomogeneous Kähler–Einstein manifolds into infinite dimensional complex projective space. It can also be regarded as an application of explicit solutions of complex Monge–Ampère equations on some pseudoconvex domains.

Light beams with general direction and polarization: Global description and geometric phase

We construct the manifold describing the family of plane monochromatic light waves with all directions, polarizations, phases and intensities. A smooth description of polarization, valid over the entire sphere S2 of directions, is given through the construction of an orthogonal basis pair of complex polarization vectors for each direction; any light beam is then uniquely and smoothly specified by giving its direction and two complex amplitudes. This implies that the space of all light beams is the six dimensional manifold S2×C2∖{0}, the (untwisted) Cartesian product of a sphere and a two dimensional complex vector space minus the origin. A Hopf map (i.e. mapping the two complex amplitudes to the Stokes parameters) then leads to the four dimensional manifold S2×S2 which describes beams with all directions and polarization states. This product of two spheres can be viewed as an ordered pair of two points on a single sphere, in contrast to earlier work in which the same system was represented using Majorana’s mapping of the states of a spin one quantum system to an unordered pair of points on a sphere. This is a different manifold, CP2, two dimensional complex projective space, which does not faithfully represent the full space of all directions and polarizations. Following the now-standard framework, we exhibit the fibre bundle whose total space is the set of all light beams of non-zero intensity, and base space S2×S2. We give the U(1) connection which determines the geometric phase as the line integral of a one-form along a closed curve in the total space. Bases are classified as globally smooth, global but singular, and local, with the last type of basis being defined only when the curve traversed by the system is given. Existing as well as new formulae for the geometric phase are presented in this overall framework.

Linear systems on a class of anticanonical rational threefolds

Let X be the blow-up of the three dimensional complex projective space along r points in very general position on a smooth elliptic quartic curve B⊂P3 and let L∈Pic(X) be any line bundle. The aim of this paper is to provide an explicit algorithm for determining the dimension of H0(X,L).

Periodic Structure of Transversal Maps on $$\mathbb{C }$$ P $$^{n}$$ , $$\mathbb{H }$$ P $$^{n}$$ and $$\mathbb{S }^{p}\times \mathbb{S }^{q}$$

A \(\mathcal{C }^{1}\) map \(f:\mathbb{M }\rightarrow \mathbb{M }\) is called transversal if for all \(m\in \mathbb{N }\) the graph of \(f^{m}\) intersects transversally the diagonal of \(M\times M\) at each point \((x,x)\) being \(x\) a fixed point of \(f^m\). Let \(\mathbb{C }\)P\(^{n}\) be the \(n\)-dimensional complex projective space, \(\mathbb{H }\)P\(^{n}\) be the \(n\)-dimensional quaternion projective space and \(\mathbb{S }^{p}\times \mathbb{S }^{q}\) be the product space of the \(p\)-dimensional with the \(q\)-dimensional spheres, \(p\ne q\). Then for the cases \(\mathbb{M }\) equal to \(\mathbb{C }\)P\(^{n}\), \(\mathbb{H }\)P\(^{n}\) and \(\mathbb{S }^{p}\times \mathbb{S }^{q}\) we study the set of periods of \(f\) by using the Lefschetz numbers for periodic points.

Isospin particle systems on quaternionic projective spaces

We construct the isospin particle system on $n$-dimensional quaternionic projective spaces in the presence of the Belavin-Polyakov-Schwarz-Tyupkin instanton by the reduction from the free particle on $(2n+1)$-dimensional complex projective space. Then we add to this system a ``quaternionic oscillator potential'' and show that this oscillatorlike system is superintegrable. We show that, besides the analogs of quadratic constants of motion of the spherical (Higgs) and $\mathbb{C}{P}^{n}$ oscillators, it possesses the third-order constants of motion, which are functionally independent from the quadratic ones.

Kähler immersions of homogeneous Kähler manifolds into complex space forms

In this paper we study the homogeneous Kaehler manifolds (h.K.m.) which can be Kaehler immersed into finite or infinite dimensional complex space forms. On the one hand we completely classify the h.K.m. which can be Kaehler immersed into a finite or infinite dimensional complex Euclidean or hyperbolic space. On the other hand, we extend known results about Kaehler immersions into the finite dimensional complex projective space to the infinite dimensional setting

Kähler–Einstein submanifolds of the infinite dimensional projective space

This paper consists of two main results. In the first one we describe all Kaehler immersions of a bounded symmetric domain into the infinite dimensional complex projective space in terms of the Wallach set of the domain. In the second one we exhibit an example of complete and non-homogeneous Kaehler-Einstein metric with negative scalar curvature which admits a Kaehler immersion into the infinite dimensional complex projective space.

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∞.

On Lagrangian submanifolds in complex hyperquadrics and isoparametric hypersurfaces in spheres

The n-dimensional complex hyperquadric is a compact complex algebraic hypersurface defined by the quadratic equation in the (n+1)-dimensional complex projective space, which is isometric to the real Grassmann manifold of oriented 2-planes and is a compact Hermitian symmetric space of rank 2. In this paper, we study geometry of compact Lagrangian submanifolds in complex hyperquadrics from the viewpoint of the theory of isoparametric hypersurfaces in spheres. From this viewpoint we provide a classification theorem of compact homogeneous Lagrangian submanifolds in complex hyperquadrics by using the moment map technique. Moreover we determine the Hamiltonian stability of compact minimal Lagrangian submanifolds embedded in complex hyperquadrics which are obtained as Gauss images of isoparametric hypersurfaces in spheres with g(= 1, 2, 3) distinct principal curvatures.

Projective coordinates and projective space limit

The “projective lightcone limit” has been proposed as an alternative holographic dual of an AdS space. It is a new type of group contraction for a coset G/H preserving the isometry group G but changing H. In contrast to the usual group contraction, which changes G preserving the spacetime dimension, it reduces the dimensions of the spacetime on which G is realized. The obtained space is a projective space on which the isometry is realized as a linear fractional transformation. We generalize and apply this limiting procedure to the “Hopf reduction” and obtain ( n − 1 )-dimensional complex projective space from ( 2 n − 1 )-dimensional sphere preserving SU( n) symmetry.