K3

K3, often written K3, refers to a class of compact complex surfaces in algebraic geometry that are distinguished by their rich geometric structure and simplicity of their canonical bundle. A K3 surface is a smooth, simply connected complex surface with trivial canonical bundle, equivalently a surface with vanishing first Chern class and a nowhere vanishing holomorphic two-form.

Topologically and Hodge-theoretically, a K3 surface has Betti numbers b0 = 1, b1 = 0, b2 = 22, b3 = 0, b4 = 1, and Hodge numbers (h^{p,q}) with h^{2,0} = h^{0,2} = 1 and h^{1,1} = 20. The second cohomology H^2(S, Z) carries a natural even, unimodular lattice structure of signature (3,19), isomorphic to II_{3,19} and often described as 3H ⊕ 2(-E8). The Picard group, or Neron-Severi group, has rank ρ between 0 and 20; generic K3 surfaces have ρ = 0, while algebraic K3 surfaces satisfy ρ ≥ 1.

Explicit constructions include smooth quartic surfaces in projective 3-space, which are K3 by adjunction, and complete intersections in projective spaces. Kummer surfaces, obtained by resolving the singularities of the quotient of an abelian surface by the involution, also yield K3 surfaces.

K3 surfaces form a 20-dimensional moduli space of complex structures, and they obey a version of the Torelli theorem: the isomorphism class is controlled by the Hodge structure on H^2 together with the lattice data. In differential geometry, K3 surfaces are examples of compact hyperkähler manifolds with holonomy SU(2).

The name K3 was coined by André Weil in honor of Kummer, Kähler, and Kodaira. K3 surfaces are central in algebraic geometry and mathematical physics due to their rich geometry, moduli, and symmetry properties.