Complex projective space

Complex projective space
Original: Mark.Howison at English Wikipedia This version: CheChe · CC BY-SA 4.0 · source
Name	Complex projective space
Field	Complex numbers

Complex projective space is a fundamental construction in modern mathematics that arises in the study of Carl Friedrich Gauss, Bernhard Riemann, Augustin-Louis Cauchy, Riemann mapping theorem contexts and in the work of Henri Poincaré, Élie Cartan, Henri Lefschetz and Jean-Pierre Serre. It connects ideas from David Hilbert's program, Emmy Noether's algebraic concepts, Alexander Grothendieck's schemes, and the differential frameworks developed by Shiing-Shen Chern and Armand Borel.

Definition and basic properties

The space is defined as the set of one-dimensional linear subspaces of a complex vector space, a construction central to Georg Cantor's set theory influences on topology and to Felix Klein's Erlangen program. It is a compact, simply connected, complex manifold studied by Élie Cartan and Weyl family researchers, playing roles in the works of André Weil, Oscar Zariski, Jean-Louis Koszul, and Hermann Weyl. The projective classification relates to David Mumford's geometric invariant theory and to moduli problems treated by Pierre Deligne, Michael Atiyah, Raoul Bott, and Shing-Tung Yau. As a homogeneous space, it admits transitive actions by SU(n+1), which were investigated by Élie Cartan and Hermann Weyl.

Coordinate charts and homogeneous coordinates

Homogeneous coordinates were popularized through the frameworks of Gaspard Monge and later formalized in algebraic geometry by Jean-Baptiste Joseph Fourier-inspired algebraists and by Alexander Grothendieck's students such as Pierre Deligne. Local coordinate charts are akin to affine pieces used by Oscar Zariski and André Weil in defining schemes and by David Mumford in geometric invariant theory. Transition maps between charts reflect change-of-basis matrices studied by Évariste Galois-lineage group theorists and appear in representation-theoretic analyses by Hermann Weyl and Emmy Noether.

Topology and differential structure

Topological properties, including compactness and simple connectedness, are classical results connected to the work of Henri Poincaré and refined by Luitzen Egbertus Jan Brouwer and Maurice Fréchet in early 20th-century topology. The differentiable manifold structure invites study via tangent bundles and vector fields as in the contributions of Shiing-Shen Chern, Raoul Bott, Michael Atiyah, and John Milnor. Morse-theoretic perspectives connect to Marston Morse and to advances in global analysis by Atiyah–Bott-style researchers and to index theorem developments by Atiyah and Isadore Singer.

Complex and Kähler geometry

The natural complex structure and the Fubini–Study metric relate to Ludwig Faddeev-adjacent studies, to Kähler geometry developed by Eugenio Calabi and studied extensively by Shing-Tung Yau, and to Ricci-flat considerations in Calabi conjecture contexts. These structures underpin links to moduli problems addressed by Donaldson and Thomas Yau-related work and to stability concepts from Kempf–Ness theorem-style results involving Simon Donaldson and Karen Uhlenbeck.

Algebraic and projective varieties

Projective varieties are defined as zero loci of homogeneous polynomials, a perspective central to David Hilbert's Nullstellensatz and expanded by Oscar Zariski and André Weil. The embedding theorems of projective geometry were refined by Federigo Enriques-lineage geometers and used by Alexander Grothendieck in scheme theory. Intersection theory within projective space owes foundational developments to Jean-Pierre Serre, René Thom-adjacent topology, and William Fulton's intersection-theoretic formalism. Moduli spaces realized as projective varieties appear in the works of Mumford, Deligne, Mark Green, and Phillip Griffiths.

Cohomology and characteristic classes

Cohomological computations involve calculations first systematized by Elie Cartan and later extended by Jean Leray, Hyman Bass-adjacent homological algebra, and by Grothendieck's spectral sequence techniques. The integral cohomology ring and Chern classes connect to the work of Shiing-Shen Chern, Raoul Bott, Michael Atiyah, and Isadore Singer via index theory and characteristic class theory. Applications to enumerative geometry and Schubert calculus reflect the influence of Hermann Schubert, Bernard Riemann-inspired enumerative traditions, and modern treatments by William Fulton and Jan Stieberger-style combinatorial geometers.

Examples and applications

Low-dimensional cases include the complex projective line, which relates historically to Bernhard Riemann's study of Riemann surfaces and to Felix Klein's work on automorphic functions, and the complex projective plane which features in classification problems explored by Federigo Enriques, Kunihiko Kodaira, and Shigefumi Mori. Applications span from classical algebraic curves in Carl Gustav Jacob Jacobi-lineage work to modern string-theoretic compactifications studied by Edward Witten, Cumrun Vafa, Andrew Strominger, and Shing-Tung Yau, and to geometric representation theory themes pursued by George Lusztig and Nicholas M. Katz. Computational approaches draw on algorithmic algebraic geometry advanced by Bernd Sturmfels, David Cox, and Bertrand Pryor-adjacent researchers, with implications for enumerative problems in the traditions of Alexander Grothendieck and Jean-Pierre Serre.

Category:Complex manifolds