complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space...

is the two-dimensional complex projective space. It is a complex manifold of complex dimension 2, described by three complex coordinates ( Z 1 , Z 2...

point", which is subject to the axioms of projective geometry. For some such set of axioms, the projective spaces that are defined have been shown to be...

In algebraic geometry, a projective variety is an algebraic variety that is a closed subvariety of a projective space. That is, it is the zero-locus in...

the Universal coefficient theorem. Complex projective space Quaternionic projective space Lens space Real projective plane See the table of Don Davis for...

simplest complex manifolds. In projective geometry, the sphere is an example of a complex projective space and can be thought of as the complex projective line...

quantum mechanics, the projective Hilbert space or ray space P ( H ) {\displaystyle \mathbf {P} (H)} of a complex Hilbert space H {\displaystyle H} is...

mathematics, quaternionic projective space is an extension of the ideas of real projective space and complex projective space, to the case where coordinates...

compared to elementary Euclidean geometry, projective geometry has a different setting (projective space) and a selective set of basic geometric concepts...

the complex vector space C n + 1 {\displaystyle \mathbb {C} ^{n+1}} . The projective model of the complex hyperbolic space is the projectivized space of...

fibers over real projective space RPn with fiber S0. The Hopf construction gives circle bundles p : S2n+1 → CPn over complex projective space. This is actually...

are holomorphic Complex projective space, a projective space with respect to the field of complex numbers Unitary space, a vector space with the addition...

isometry group of complex projective space, just as the projective orthogonal group is the isometry group of real projective space. In terms of matrices...

if a CW-complex has no cells in consecutive dimensions, then all of its homology modules are free. For example, the complex projective space C P n {\displaystyle...

{\displaystyle \mathbb {Z} [x]/x^{n+1}.} In the case of infinite complex projective space, taking limits gives the answer Z [ x ] . {\displaystyle \mathbb...

otherwise. A projective complex analytic variety is a subset X ⊆ C P n {\displaystyle X\subseteq \mathbb {CP} ^{n}} of complex projective space that is, in...

in which the hairs are complex (see the example of the complex hairy ball theorem below), or for 1-dimensional projective spaces over many other fields...

analogous way, the complex projective space C P ∞ {\displaystyle \mathbb {C} \mathbf {P} ^{\infty }} carries a universal complex line bundle. In this...

that they can represent "holes" in a topological space. However, homotopy groups are often very complex and hard to compute. In contrast, homology groups...

algebraic geometry, the other being projective geometry. A complex affine space can be obtained from a complex projective space by fixing a hyperplane, which...

especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action...

called the projective plane; the qualifier "real" is added to distinguish it from other projective planes such as the complex projective plane and finite...

contained in U.) Complex dynamics has been effectively developed in any dimension. This section focuses on the mappings from complex projective space C P n {\displaystyle...

projective space Pn is a moduli space that parametrizes the space of lines in Rn+1 which pass through the origin. Similarly, complex projective space...

sheaf cohomology groups on complex projective space. The projective space in question is the twistor space, a geometrical space naturally associated to the...

connected symmetric spaces. (For example, the universal cover of a real projective plane is a sphere.) Second, the product of symmetric spaces is symmetric,...

} The infinite dimensional complex projective space C P ∞ {\displaystyle \mathbb {CP} ^{\infty }} is the classifying space BS1 for the circle S1 thought...

Examples hyperbolic space Gauss–Bolyai–Lobachevsky space Grassmannian Complex projective space Real projective space Euclidean space Stiefel manifold Upper...

complex manifolds. For example, on a compact complex manifold X {\displaystyle X} (like complex projective space or the vanishing locus in projective...

that cannot be embedded in projective space and are not algebraic. Analogy of the Levi problems on the complex projective space C P n {\displaystyle \mathbb...