K3 Surface

A complex analytic K3 surface is a compact connected complex manifold of dimension two with a trivial canonical bundle and irregularity zero; an algebraic K3 surface over any field is a smooth, proper, geometrically connected algebraic surface satisfying the same conditions. In the Enriques-Kodaira classification of surfaces, K3 surfaces form one of the four classes of minimal surfaces of Kodaira dimension zero, and a simple example is the Fermat quartic surface defined in complex projective three-space. Together with two-dimensional compact complex tori, K3 surfaces are the Calabi-Yau manifolds, and also the hyperkahler manifolds, of dimension two, placing them at the center of the classification of algebraic surfaces, between the positively curved del Pezzo surfaces, which are easy to classify, and the negatively curved surfaces of general type, which are essentially unclassifiable. K3 surfaces can be considered the simplest algebraic varieties whose structure does not reduce to curves or abelian varieties while still admitting substantial understanding, and a complex K3 surface, of real dimension four, plays an important role in the study of smooth four-manifolds. K3 surfaces have been applied to Kac-Moody algebras, mirror symmetry and string theory.