CP^n

CP^n
Name	CP^n
Type	Complex projective space
Field	Complex numbers
Notable	Fubini–Study metric, Hopf fibration

CP^n

Complex projective n-space is the space of complex lines through the origin in C^{n+1}, a compact complex manifold and a fundamental example in Algebraic Geometry, Differential Geometry, and Topology. It appears throughout the work of figures such as Henri Poincaré, Élie Cartan, Claude Chevalley, Kunihiko Kodaira, and Alexander Grothendieck and plays a central role in results like the Riemann–Roch theorem, the Hirzebruch–Riemann–Roch theorem, and applications by William Fulton and Joe Harris.

Definition and basic properties

CP^n is defined as the quotient of C^{n+1}\{0} by the C^× action, giving a compact, simply connected complex manifold with complex dimension n; classical constructions involve the Hopf fibration studied by Heinz Hopf and links to the S^2n+1 sphere examined by John Milnor and Raoul Bott. As a homogeneous space it can be realized as U(n+1)/(U(1)×U(n)) in the lineage of work by Élie Cartan and Hermann Weyl, and carries a natural Fubini–Study form studied in the context of the Calabi conjecture proven by Shing-Tung Yau. CP^n admits holomorphic embeddings of projective varieties used in the proofs by David Mumford and Alexander Grothendieck in Geometric Invariant Theory.

Complex projective coordinates and charts

Projective coordinates [z_0:...:z_n] originate in classical treatments by Jean-Victor Poncelet and appear in modern expositions by Robin Hartshorne and Phillip Griffiths; affine charts U_i = {z_i ≠ 0} are biholomorphic to C^n, a technique used by Kunihiko Kodaira and Shoshichi Kobayashi for local studies. Transition maps between charts reflect coordinate changes central to the analyses by Charles B. Morrey and Shirley Temple (mathematics) — see algebraic constructions as in texts by Joe Harris and William Fulton. Homogeneous coordinates also underpin projective embeddings such as the Veronese map employed by Francesco Severi and the Segre embedding used by David Hilbert.

Topology and homology

The integral homology and cohomology rings of CP^n were calculated in classical topology by L. E. J. Brouwer and refined by Hassler Whitney and John Milnor; H^*(CP^n;Z) ≅ Z[α]/(α^{n+1}) with generator α in degree 2, a structure appearing in the work of Raoul Bott on periodicity and in computations by Serre and Henri Cartan. The cell decomposition by complex projective cells parallels constructions in Poincaré duality used by Emmy Noether and Jean Leray and appears in intersection theory developed by William Fulton. Characteristic classes such as Chern classes of the tangent bundle were studied by Shiing-Shen Chern and feed into index theorems by Atiyah–Singer and applications by Michael Atiyah and Isadore Singer.

Complex and Kähler geometry

CP^n carries the Fubini–Study Kähler metric central to work by Eugenio Calabi and Shing-Tung Yau; positivity properties of its curvature are used in proofs by S.-T. Yau and in stability discussions by Simon Donaldson and Shing-Tung Yau. Holomorphic sectional curvature computations appear in the research of Kunihiko Kodaira and Shoshichi Kobayashi and feed into rigidity results explored by André Weil and Hermann Weyl. CP^n provides prototypical examples in Kähler–Einstein theory, geometric quantization treatments by Bertram Kostant and Jean-Marie Souriau, and moment-map techniques by Mumford with implications for the Yang–Mills equations studied by Karen Uhlenbeck.

Line bundles and sheaf cohomology

The hyperplane line bundle O(1) and its powers O(k) are standard objects introduced in classical projective investigations by Alexander Grothendieck and used in cohomology computations in texts by Robin Hartshorne; cohomology groups H^i(CP^n,O(k)) follow the Bott formula as elaborated by Raoul Bott and Hans Samelson. Serre duality and Kodaira vanishing theorems proved by Jean-Pierre Serre and Kunihiko Kodaira control these groups, and the use of sheaf cohomology in proving embedding theorems links to the work of André Weil and Oscar Zariski. Line bundle positivity notions trace to Jean-Pierre Demailly and figure in moduli problems addressed by David Mumford.

Applications and examples

CP^n hosts projective varieties central to Algebraic Geometry such as hypersurfaces studied by Bernhard Riemann, Alexander Grothendieck, and Oscar Zariski, including Calabi–Yau hypersurfaces used in string-theory models by Edward Witten and Cumrun Vafa. Complex projective curves (Riemann surfaces) embed into CP^2 in constructions by Bernhard Riemann and Henri Poincaré; Grassmannians and flag varieties relate CP^n to works by Ehresmann and Weyl and appear in representation theory of Élie Cartan and Weyl. CP^n serves as a testbed in symplectic topology via Gromov–Witten invariants developed by Mikhail Gromov and Yuri Manin, and in mathematical physics in instanton and monopole moduli explored by Nigel Hitchin and Edward Witten.

Category:Complex manifolds