N-dimensional Complex Projective Space Research Articles

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>

Almost complex structures on connected sums of almost complex manifolds and complex projective spaces

In this paper, the existence of almost complex structures on connected sums of almost complex manifolds and complex projective spaces are investigated. Firstly, we show that if M is a 2n-dimensional almost complex manifold, then so is $$M\sharp \overline{{\mathbb {C}}P^{n}}$$ , where $$\overline{{\mathbb {C}}P^{n}}$$ is the n-dimensional complex projective space with the reversed orientation. Secondly, for any positive integer $$\alpha $$ and any 4n-dimensional almost complex manifolds $$M_{i}, ~1\le i \le \alpha $$ , we prove that $$\left( \sharp _{i=1}^{\alpha } M_{i}\right) \sharp (\alpha {-}1) {\mathbb {C}}P^{2n}$$ must admit an almost complex structure. At last, as an application, we obtain that $$\alpha {\mathbb {C}}P^{2n}\sharp ~\beta \overline{{\mathbb {C}}P^{2n}}$$ admits an almost complex structure if and only if $$\alpha $$ is odd.

Periods of continuous maps on some compact spaces

The objective of this paper is to provide information on the set of periodic points of a continuous self-map defined in the following compact spaces: (the n-dimensional sphere), (the product space of the n-dimensional with the m-dimensional spheres), (the n-dimensional complex projective space) and (the n-dimensional quaternion projective space). We use as main tool the action of the map on the homology groups of these compact spaces.

On parallelizability and span of the Dold manifolds

The Dold manifold P (m,n) is obtained from the product Sm × CPn of the m-dimensional sphere and n-dimensional complex projective space by identifying (x, [z1, . . . , zn+1]) with (−x, [z1, . . . , zn+1]), where z denotes the complex conjugate of z. We answer the parallelizability question for the Dold manifolds P (m,n) and, by completing an earlier (2008) result due to Peter Novotný, we solve the vector field problem for the manifolds P (m, 1).

Global and local definition of the Monge–Ampère operator on compact Kähler manifolds

The aim of this Note is to give a sufficient condition in order for a function in the global domain of definition of the Monge–Ampère operator not to belong to the local domain of the former in the sense of Cegrell, when one looks at the n-dimensional complex projective space. Using this result, we show that the subsolution theorem is false for functions in the local domain of definition of the Monge–Ampère operator on such a projective space.

On the Lefschetz periodic point free continuous self-maps on connected compact manifolds

We characterize the Lefschetz periodic point free self-continuous maps on the following connected compact manifolds: C P n the n-dimensional complex projective space, H P n the n-dimensional quaternion projective space, S n the n-dimensional sphere and S p × S q the product space of the p-dimensional with the q-dimensional spheres.

Landau's Theorem for Holomorphic Curves in Projective Space and The Kobayashi Metric on Hyperplane Complements

We prove an effective version of a theorem of Dufresnoy: For any set of 2n+1 hyperplanes in general position in n-dimensional complex projective space, we find an explicit constant K such that for every holomorphic map f from the unit disc to the complement of these hyperplanes, the derivative of f at the origin measured with respect to the Fubuni-Study metric is bouned above by K. This result gives an explicit lower bound on the Royden function, i.e., the ratio of the Kobayashi metric on the hyperplane complement to the Fubini-Study metric. Our estimate is based on the potential-theoretic method of Eremenko and Sodin.

On the existence of singular directions of holomorphic maps from the unit disk into P n(C)

This article proves the existence of Julia directions of value distribution of holomorphic mapping f from the unit disk into the n-dimensional complex projective space P n (C) under the assumption lim sup r → 1 - T ( r , f ) / log 1 1 - r = + ∞ for hypersurfaces in general position. A heuristic principle concerning the existence of Julia directions of holomorphic mappings from the unit disk into P n (C) is given also.

Action and index spectra and periodic orbits in Hamiltonian dynamics

The main theme of this paper is the connection between the existence of infinitely many periodic orbits for a Hamiltonian system and the behavior of its action or index spectrum under iterations. We use the action and index spectra to show that any Hamiltonian diffeomorphism of a closed, rational manifold with zero first Chern class has infinitely many periodic orbits and that, for a general rational manifold, the number of geometrically distinct periodic orbits is bounded from below by the ratio of the minimal Chern number and half of the dimension. These generalizations of the Conley conjecture follow from another result proved here asserting that a Hamiltonian diffeomorphism with a symplectically degenerate maximum on a closed rational manifold has infinitely many periodic orbits. We also show that for a broad class of manifolds and/or Hamiltonian diffeomorphisms the minimal action--index gap remains bounded for some infinite sequence of iterations and, as a consequence, whenever a Hamiltonian diffeomorphism has finitely many periodic orbits, the actions and mean indices of these orbits must satisfy a certain relation. Furthermore, for Hamiltonian diffeomorphisms of the n-dimensional complex projective space with exactly n+1 periodic orbits a stronger result holds. Namely, for such a Hamiltonian diffeomorphism, the difference of the action and the mean index on a periodic orbit is independent of the orbit, provided that the symplectic structure on the projective space is normalized to be in the same cohomology class as the first Chern class.

Lower order eigenvalues of Dirichlet Laplacian

In this paper, we investigate an eigenvalue problem for the Dirichlet Laplacian on a domain in an n-dimensional compact Riemannian manifold. First we give a general inequality for eigenvalues. As one of its applications, we study eigenvalues of the Laplacian on a domain in an n-dimensional complex projective space, on a compact complex submanifold in complex projective space and on the unit sphere. By making use of the orthogonalization of Gram–Schmidt (QR-factorization theorem), we construct trial functions. By means of these trial functions, estimates for lower order eigenvalues are obtained.

The quantum oscillator on complex projective space (Lobachewski space) in a constant magnetic field and the issue of generic boundary conditions

We perform a one-parameter family of self-adjoint extensions characterized by the parameter ω0. This allows us to get generic boundary conditions for the quantum oscillator on N-dimensional complex projective space and on its non-compact version, i.e., Lobachewski space in the presence of a constant magnetic field. As a result, we get a family of energy spectra for the oscillator. In our formulation the already known result of this oscillator also belongs to the family. We have also obtained an energy spectrum which preserves all the symmetries (full-hidden symmetry and rotational symmetry) of the oscillator. The method of self-adjoint extensions has also been discussed for a conic oscillator in the presence of the constant magnetic field.

ON THE SINGULAR DIRECTIONS OF VALUE DISTRIBUTION OF HOLOMORPHIC CURVES IN P n( C)

This article proves the existence of singular directions of value distribution theory for some transcendental holomorphic curves in the n-dimensional complex projective space P n ( C). An example is given to complement these results.

Homotopy classification of twisted complex projective spaces of dimension 4

We study the existence problem of a 2 n dimensional Poincare complex whose homology is isomorphic to that of n dimensional complex projective space when n = 4 .

SOME EXAMPLES OF HYPERBOLIC HYPERSURFACES IN THE COMPLEX PROJECTIVE SPACE

In the previous paper [6], the author constructed hyperbolic hypersurfaces of degree <TEX>$2^{n}$</TEX> in the n-dimensional complex projective space for every <TEX>$n\;\geq\;3$</TEX>. The purpose of this paper is to give some improvement of this result and to show some general methods of constructions of hyperbolic hypersurfaces of higher degree, which enable us to construct hyperbolic hypersurfaces of degree d in the n-dimensional complex projective space for every <TEX>$d\;\geq\;2\;{\times}\;6^{n}$</TEX>.

Radon and Fourier transforms for [formula omitted]-modules

The Fourier and Radon hyperplane transforms are closely related, and one such relation was established by Brylinski [4] in the framework of holonomic D-modules. The integral kernel of the Radon hyperplane transform is associated with the hypersurface SCP P of pairs ðx; yÞ; where x is a point in the n-dimensional complex projective space P belonging to the hyperplane yAP : As it turns out, a useful variant is obtained by considering the integral transform associated with the open complement U of S in P P : In the first part of this paper, we generalize Brylinski’s result in order to encompass this variant of the Radon transform, and also to treat arbitrary quasi-coherent D-modules, as well as (twisted) abelian sheaves. Our proof is entirely geometrical, and consists in a reduction to the onedimensional case by the use of homogeneous coordinates. The second part of this paper applies the above result to the quantization of the Radon transform, in the sense of [7]. First we deal with line bundles. More precisely, let P 1⁄4 PðVÞ be the projective space of lines in the vector spaceV; denote by ð Þ 3 D R

Integrable anharmonic oscillators on spheres and hyperbolic spaces

We construct two new examples of integrable Hamiltonian systems. They describe the motion of a particle under the influence of certain quartic potentials. The first system describes such a motion on the n-dimensional sphere. The second describes the motion on the n-dimensional hyperbolic space. Here n is an arbitrary positive integer. We represent the system on a (2n + 1)-dimensional sphere with an additional U(1)-symmetry as a symplectic reconstruction of a system which has a topologically non-trivial magnetic term and whose configuration space is the n-dimensional complex projective space. We use this description to give an alternative proof of the integrability of the system on the odd-dimensional sphere.

A generalization of picard's theorem with moving targets

For given 2n+1 hyperplanes with moving targets in pointwise general position and any holomorphic map f from into a n-dimensional complex projective space omitting those 2n+1 hyperplanes, we show that there exist finitely many (n+1)×(n+1) everywhere invertible matrices with entire holomorphic functions as entries such that the holomorphic map f multiplied by one of the matrices becomes constant. In addition, this set of matrices depends only on the hyperplanes, and can be determined effectively.

Numerical properties of Chern classes of certain covering manifolds

In this paper, we study numerical properties of Chern classes of certain covering manifolds. One of the main results is the following: Let ψ : X → P n be a finite covering of the n-dimensional complex projective space branched along a hypersurface with only simple normal crossings and suppose X is nonsingular. Let c i ( X) be the i-th Chern class of X. Then (i) if the canonical divisor K X is numerically effective, then (−1) k c k ( X) ( k ≥ 2) is numerically positive, and (ii) if X is of general type, then (−1) n c i l ( X) ⋯ c i r , ( X) > 0, where i l + … + i r = n. Furthermore we show that the same properties hold for certain Kummer coverings.

N-dimensional totally real minimal submanifolds of CP n

Let CPn be the n-dimensional complex projective space with the Study-Fubini metric of constant holomorphic sectional curvature 4 and let M be a compact, orientable, n-dimensional totally real minimal submanifold of CPn. In this paper we prove the following results. (a) If M is 6-dimensional, conformally flat and has non negative Euler number and constant scalar curvature τ, 0<τ ≦ 70/3, then M is locally isometric to S1,5 :=S1 (sin θ cos θ) × S5 (sin θ), tan θ = √6. (b) If M is 4-dimensional, has parallel second fundamental form and scalar curvature τ ≧ 15/2, then M is locally isometric to S1,3 :=S1 (sin θ cos θ) × S3 (sinθ), tan θ=2, or it is totally geodesic.

Circles in a complex projective space

The study of circles is one of the interesting objects in differential geometry. A curve γ(s) on a Riemannian manifold M parametrized by its arc length 5 is called a circle, if there exists a field of unit vectors Ys along the curve which satisfies, together with the unit tangent vectors Xs — Ϋ(s)9 the differential equations : FSXS = kYs and FsYs= — kXs, where k is a positive constant, which is called the curvature of the circle γ(s) and Vs denotes the covariant differentiation along γ(s) with respect to the Aiemannian connection V of M. For given a point lEJIί, orthonormal pair of vectors u, v^ TXM and for any given positive constant k, we have a unique circle γ(s) such that γ(0)=x, γ(0) = u and (Psγ(s))s=o=kv. It is known that in a complete Riemannian manifold every circle can be defined for o o < 5 < o o (Cf. [6]). The study of global behaviours of circles is very interesting. However there are few results in this direction except for the global existence theorem. In general, a circle in a Riemannian manifold is not closed. Here we call a circle γ(s) closed if there exists So with 7(so) = /(0), XSo Xo and YSo— Yo. Of course, any circles in Euclidean m-space E are closed. And also any circles in Euclidean m-sphere S(c) are closed. But in the case of a real hyperbolic space H(c), there exist many open circles. It is well-known that in H(c) circles with curvature not exceeding v |c | are open and circles with curvature greater than v \c are closed (cf. [3]). That is, in H{c) the answer to the question Is a circle γ(s) closed? depends on its curvature. In this paper, we are concerned with circles in an n-dimensional complex projective space CP(c) of constant holomorphic sectional curvature c. In Section 1, by using Submanifold Theory we give an interesting family of open circles and closed circles with the same curvature /c/2^ in CP(c). In his paper [5], Naitoh