CP^2

CP^2
Name	CP^2
Type	Complex projective plane
Dimension	2 (complex)
Homology	Z, 0, Z, 0, Z

CP^2 CP^2 is the complex projective plane, a compact two-dimensional complex manifold and a fundamental example in algebraic geometry, differential topology, and mathematical physics. It connects to classical objects such as Riemann sphere, Projective space, Grassmannian manifold, Fubini–Study metric, and appears in studies by figures like Hodge, Kodaira, Atiyah, and Donaldson.

Definition and basic properties

CP^2 is defined as the set of complex lines through the origin in C^3, realized by homogeneous coordinates and quotient constructions that relate to GL(3,C), PGL(3,C), SL(3,C), and actions of U(3). As a smooth manifold CP^2 is simply connected and closed, appearing alongside classical examples such as S^4, S^2 × S^2, K3 surface, and Complex torus in classification results by Freedman, Donaldson, and Seiberg–Witten theory. CP^2 admits an ample hyperplane class coming from projective embeddings studied by Noether, Castelnuovo, and Mori.

Complex projective coordinates and charts

CP^2 is covered by three standard affine charts obtained by setting one homogeneous coordinate nonzero, akin to constructions in Projective geometry, Affine variety, and coordinates used by Weierstrass, Grothendieck, and Serre. Homogeneous coordinates [z0:z1:z2] identify points up to scalar multiplication by elements of C^×, a construction related to quotients by C^* and moduli descriptions familiar in work of Mumford and Geometric invariant theory. Transition maps between charts are rational maps much like those in studies by Cremona, Hirzebruch, and Segre.

Topology and homology

The integral homology groups of CP^2 are concentrated in even degrees, with H0 ≅ Z, H2 ≅ Z, H4 ≅ Z, reflecting intersection forms studied by Intersection form, Poincaré duality, and calculations by Poincaré, Lefschetz, and Hirzebruch in relation to the Todd class and signature theorems of Atiyah–Singer. The generator of H2 corresponds to the projective line class, which interacts with Chern classes appearing in theorems of Chern, Gauss–Bonnet, and Hirzebruch–Riemann–Roch. CP^2's cup product structure and cohomology ring are prototypes in examples considered by Bott, Samelson, and Milnor.

Symplectic and complex geometry

CP^2 carries a canonical Kähler structure linking symplectic forms studied by Moser, Gromov, and McDuff with complex structures central to work by Kodaira and Newlander–Nirenberg. The Fubini–Study form yields monotone symplectic properties invoked in pseudoholomorphic curve techniques introduced by Gromov and used in quantum cohomology by Kontsevich and Witten. Symplectic packing and exceptional curve analyses reference results by Biran, Polterovich, and Taubes.

Metric and curvature (Fubini–Study metric)

The Fubini–Study metric on CP^2 is the unique up-to-scale U(3)-invariant Kähler–Einstein metric, studied classically in works by Cartan, Calabi, and Aubin. Its sectional curvature ranges between positive bounds and is used in comparison theorems by Bonnet–Myers and rigidity statements by Berger and Matsushima. The metric underlies analyses in spectral geometry by Hodge and heat kernel techniques by Atiyah–Bott.

Holomorphic line bundles and sheaf cohomology

Holomorphic line bundles on CP^2 are classified by degree and generated by the hyperplane bundle O(1), central to Serre duality, cohomological vanishing theorems of Kodaira, and the Borel–Weil theorem studied by Borel and Weil. Cohomology groups H^i(O(k)) are computed by classical methods of Serre, Grothendieck, and Hirzebruch–Riemann–Roch and are foundational in constructions by Hartshorne and projective embedding techniques used by Castelnuovo–Mumford regularity and Bertini theorems.

Applications and examples (> embeddings, instantons)

CP^2 appears as the target or moduli space in many contexts: projective embeddings of algebraic surfaces studied by Enriques, Kodaira–Spencer, and Mumford; twistor constructions relating to Penrose and instanton moduli related to Atiyah–Drinfeld–Hitchin–Manin and work by Atiyah and Hitchin; Yang–Mills instantons and anti-self-dual connections explored by Donaldson, Uhlenbeck, and Taubes; examples in mirror symmetry and quantum cohomology by Kontsevich and Givental; and appearing in classification problems linked to Wall, Freedman, and exotic smooth structures studied by Kronheimer.

Category:Complex manifolds