complex projective space

complex projective space
Name	Complex projective space
Caption	A visualization of the complex projective plane, often represented via a gluing (topology) of a 3-sphere.
Field	Algebraic geometry, Topology, Complex geometry
Properties	Compact, Simply connected, Kähler manifold

complex projective space is a fundamental object in mathematics, serving as a central example in algebraic geometry, differential geometry, and topology. It is the space of all complex lines passing through the origin in complex coordinate space of one higher dimension. First studied systematically by Georges de Rham and Solomon Lefschetz, its properties are deeply intertwined with homogeneous coordinates introduced by August Ferdinand Möbius and later developed by Arthur Cayley and Felix Klein.

Definition and construction

The standard construction begins with the complex vector space ℂⁿ⁺¹ \ {0}, considering the equivalence relation where two points are identified if one is a complex scalar multiple of the other. The resulting quotient space under this group action by the multiplicative group ℂ* is denoted ℂPⁿ. This construction parallels that of the real projective space but with complex coefficients, and it naturally arises in the study of projective geometry initiated by Jean-Victor Poncelet. The space can also be realized as a compactification of ℂⁿ, similar to how the Riemann sphere compactifies the complex plane.

Homogeneous coordinates

Points are described using homogeneous coordinates, written as [Z₀ : Z₁ : ... : Z_n], where the coordinates are not all zero and are defined up to multiplication by a non-zero complex number. These coordinates were fundamental in the work of Julius Plücker and are essential for defining projective varieties. The coordinate charts are given by setting one coordinate to 1, providing an atlas that shows the space is a complex manifold. This system is crucial in classical algebraic geometry, as employed by David Hilbert in his study of invariant theory.

Topology and geometry

As a topological space, it is compact and simply connected, with a cell complex structure consisting of one cell in each even dimension. This decomposition, related to the CW complex structure, was elucidated by Heinz Hopf in his work on the Hopf fibration, which describes ℂP¹ as the 2-sphere. Geometrically, it carries a canonical Fubini–Study metric, making it a Kähler manifold and an example of a Hermitian symmetric space. Its curvature properties were studied extensively by Élie Cartan and Shiing-Shen Chern.

Algebraic geometry perspective

In algebraic geometry, it is the simplest example of a projective variety and serves as the ambient space for many others, following the foundational work of Oscar Zariski and André Weil. The sheaf cohomology of its line bundles, particularly the tautological line bundle, is computed using Čech cohomology. Its subvarieties are defined by homogeneous polynomials, a central theme in the Italian school of algebraic geometry led by Francesco Severi and Guido Castelnuovo. The Chow ring of the space, describing algebraic cycles, was formalized by William Fulton.

Cohomology and characteristic classes

The cohomology ring with integer coefficients is a truncated polynomial ring ℤ[H]/(Hⁿ⁺¹), where H is the Poincaré dual of a hyperplane class. This structure was determined using tools from homotopy theory developed by Jean-Pierre Serre and René Thom. The total Chern class of its tangent bundle is (1+H)ⁿ⁺¹, a result closely associated with Michael Atiyah and Friedrich Hirzebruch in the context of the Atiyah–Singer index theorem. Its Pontryagin classes are also expressible in terms of H, relevant to work by John Milnor on exotic spheres.

Applications

It appears in numerous branches of theoretical physics, notably in quantum mechanics where the state space of a quantum system is a projective Hilbert space, an idea traceable to Paul Dirac and John von Neumann. In string theory, it is used in the compactification of extra dimensions, as explored in Calabi–Yau manifold research by Shing-Tung Yau. Within algebraic topology, it provides key examples for the study of vector bundles and obstruction theory, fields advanced by Raoul Bott and J. Frank Adams. Its finite-dimensional analogues are studied in incidence geometry and combinatorial design theory.

Category:Projective geometry Category:Complex manifolds Category:Algebraic varieties