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.

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.

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 Critical hypersurfaces and instability for reconstruction of scenes in high dimensional projective spaces

2020 ◽  
Vol 29 (1) ◽  
pp. 3-20
Author(s):  
Marina Bertolini ◽  
Luca Magri
Keyword(s):  
Projective Space ◽  
Projective Spaces ◽  
High Dimensional ◽  
Projective Reconstruction ◽  
Algebraic Varieties ◽  
Reconstruction Problem ◽  
Complex Projective Spaces ◽  
Simulated Experiment ◽  
Complex Projective ◽  
Practical Implications

In the context of multiple view geometry, images of static scenes are modeled as linear projections from a projective space P^3 to a projective plane P^2 and, similarly, videos or images of suitable dynamic or segmented scenes can be modeled as linear projections from P^k to P^h, with k>h>=2. In those settings, the projective reconstruction of a scene consists in recovering the position of the projected objects and the projections themselves from their images, after identifying many enough correspondences between the images. A critical locus for the reconstruction problem is a configuration of points and of centers of projections, in the ambient space, where the reconstruction of a scene fails. Critical loci turn out to be suitable algebraic varieties. In this paper we investigate those critical loci which are hypersurfaces in high dimension complex projective spaces, and we determine their equations. Moreover, to give evidence of some practical implications of the existence of these critical loci, we perform a simulated experiment to test the instability phenomena for the reconstruction of a scene, near a critical hypersurface.

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.

Rational Homotopy Types with the Rational Cohomology Algebra of Stunted Complex Projective Space

1992 ◽  
Vol 44 (6) ◽  
pp. 1241-1261 ◽  
Author(s):  
Gregory Lupton ◽  
Ronald Umble
Keyword(s):  
Projective Space ◽  
Spectral Sequence ◽  
Complex Projective Space ◽  
Classification Theory ◽  
Cohomology Algebra ◽  
Homotopy Equivalence ◽  
Rational Homotopy ◽  
Homotopy Types ◽  
Rational Cohomology ◽  
Complex Projective

AbstractWe consider the number of spaces, up to rational homotopy equivalence, which have rational cohomology algebra isomorphic to that of stunted complex projective space . Using a classification theory due to Schlessinger and Stasheff, we determine the number of rational homotopy types with rational comology algebra isomorphic to , for any given n and k. The necessary computations make use of a spectral sequence introduced by the second named author.

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