On projective spaces over two skew fields

1966 ◽  
Vol 62 (3) ◽  
pp. 395-398
Author(s):  
R. H. F. Denniston
Keyword(s):  
Projective Space ◽  
Complex Projective Space ◽  
Classical Problem ◽  
Projective Spaces ◽  
The Other ◽  
Real Projective Space ◽  
Skew Fields ◽  
Complex Projective

Introduction. It is a classical problem to construct a complex projective space, using as ‘imaginary points’ objects of some kind from the geometry of real projective space. (The various solutions to this problem have been reviewed in a book by Coolidge ((1)).) The object of the present paper is to solve the corresponding problem for n-dimensional projective spaces over two skew fields, one being a second-rank extension of the other. The method used seems not to have been published before, even in connexion with the classical problem.

scholarly journals Rigidity of complex convex divisible sets

2018 ◽  
Vol 10 (04) ◽  
pp. 817-851
Author(s):  
Andrew M. Zimmer
Keyword(s):  
Projective Space ◽  
Complex Projective Space ◽  
Discrete Group ◽  
Convex Sets ◽  
Convex Set ◽  
Real Projective Space ◽  
Open Convex ◽  
Strictly Convex ◽  
Complex Projective

An open convex set in real projective space is called divisible if there exists a discrete group of projective automorphisms which acts cocompactly. There are many examples of such sets and a theorem of Benoist implies that many of these examples are strictly convex, have [Formula: see text] boundary, and have word hyperbolic dividing group. In this paper we study a notion of convexity in complex projective space and show that the only divisible complex convex sets with [Formula: see text] boundary are the projective balls.

On the first pontrjagin class of homotopy complex projective spaces

2012 ◽  
Vol 62 (3) ◽  
Author(s):  
Yasuhiko Kitada
Keyword(s):  
Projective Space ◽  
Smooth Manifold ◽  
Complex Projective Space ◽  
Projective Spaces ◽  
Homotopy Equivalent ◽  
Complex Projective Spaces ◽  
Pontrjagin Class ◽  
Complex Projective ◽  
Homotopy Complex

AbstractLet M 2n be a closed smooth manifold homotopy equivalent to the complex projective space ℂP(n). It is known that the first Pontrjagin class p 1(M) of M 2n has the form (n+1+24α(M))u 2 for some integer α(M) where u is a generator of H 2(M; ℤ). We prove that α(M) is even when n is even but not divisible by 64.

Stable homotopy types of stunted complex projective spaces

1973 ◽  
Vol 73 (3) ◽  
pp. 431-438 ◽  
Author(s):  
S. Feder ◽  
S. Gitler
Keyword(s):  
Projective Space ◽  
Complex Projective Space ◽  
Projective Spaces ◽  
Stable Homotopy ◽  
Complex Dimension ◽  
Complex Projective Spaces ◽  
Homotopy Types ◽  
Complex Projective

Let CPn denote the complex projective space of complex dimension n. If k < n, CPk−1 is embedded as the (2k –1)-skeleton of CPn and we denote by , the stunted projective space, the quotient of CPn. by CPk−1.

scholarly journals The index of harmonic maps from surfaces to complex projective spaces

2020 ◽  
Vol 31 (09) ◽  
pp. 2050069
Author(s):  
J. Oliver
Keyword(s):  
Projective Space ◽  
Lower Bounds ◽  
Complex Projective Space ◽  
Harmonic Maps ◽  
Projective Spaces ◽  
Line Bundles ◽  
Complex Projective Spaces ◽  
Holomorphic Sections ◽  
Higher Genus ◽  
Complex Projective

We estimate the dimensions of the spaces of holomorphic sections of certain line bundles to give improved lower bounds on the index of complex isotropic harmonic maps to complex projective space from the sphere and torus, and in some cases from higher genus surfaces.

scholarly journals Koszul property for points in projective spaces

2001 ◽  
Vol 89 (2) ◽  
pp. 201 ◽  
Author(s):  
Aldo Conca ◽  
Ngô Viêt Trung ◽  
Giuseppe Valla
Keyword(s):  
Projective Space ◽  
General Position ◽  
Complex Projective Space ◽  
Projective Spaces ◽  
Free Resolution ◽  
General Linear ◽  
Coordinate Ring ◽  
Koszul Property ◽  
Complex Projective

A graded $K$-algebra $R$ is said to be Koszul if the minimal $R$-free graded resolution of $K$ is linear. In this paper we study the Koszul property of the homogeneous coordinate ring $R$ of a set of $s$ points in the complex projective space $\boldsymbol P^n$. Kempf proved that $R$ is Koszul if $s\leq 2n$ and the points are in general linear position. If the coordinates of the points are algebraically independent over $\boldsymbol Q$, then we prove that $R$ is Koszul if and only if $s\le 1 +n + n^2/4$. If $s\le 2n$ and the points are in linear general position, then we show that there exists a system of coordinates $x_0,\dots,x_n$ of $\boldsymbol P^n$ such that all the ideals $(x_0,x_1,\dots,x_i)$ with $0\le i \le n$ have a linear $R$-free resolution.

scholarly journals Hyperbolicity of the complement of arrangements of non complex lines

Filomat ◽  
2020 ◽  
Vol 34 (9) ◽  
pp. 3109-3118
Author(s):  
Fathi Haggui ◽  
Abdessami Jalled
Keyword(s):  
Projective Space ◽  
Complex Projective Space ◽  
The Other ◽  
Holomorphic Curves ◽  
Real Dimension ◽  
Kobayashi Hyperbolicity ◽  
Other Hand ◽  
Complex Lines ◽  
Complex Projective ◽  
Real Subspace

The goal of this paper is twofold. We study holomorphic curves f:C ? C3 avoiding four complex hyperplanes and a real subspace of real dimension five in C3 where we study the cases where the projection of f into the complex projective space CP2 is constant. On the other hand, we investigate the kobayashi hyperbolicity of the complement of five perturbed lines in CP2.

scholarly journals From surfaces in the 5-sphere to 3-manifolds in complex projective 3-space

2002 ◽  
Vol 66 (3) ◽  
pp. 465-475 ◽  
Author(s):  
J. Bolton ◽  
C. Scharlach ◽  
L. Vrancken
Keyword(s):  
Projective Space ◽  
Minimal Surface ◽  
Complex Projective Space ◽  
Lagrangian Submanifold ◽  
3 Dimensional ◽  
Ellipse Of Curvature ◽  
Complex Projective

In a previous paper it was shown how to associate with a Lagrangian submanifold satisfying Chen's equality in 3-dimensional complex projective space, a minimal surface in the 5-sphere with ellipse of curvature a circle. In this paper we focus on the reverse construction.

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