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# K3 surface

A K3 manifold is a hyperkähler manifold of real dimension 4, i.e. a four-dimensional connected manifold with SU(2) holonomy. All K3 manifolds can be showed to be diffeomorphic to each other. These manifolds first arose in algebraic geometry: they are important examples of complex algebraic surfaces (complex dimension 2 being real dimension 4); they received the name K3 surface, therefore, in that context — it is after three algebraic geometers, Kummer, Kähler and Kodaira, alluding also to the mountain peak K2 in the news at the time in the 1950s when the name was given. There are K3 manifolds that are not algebraic surfaces.

A number of possible characterisations can be given, in geometric terms. A non-singular quartic surface in projective space of three dimensions is a K3 surface. This determines the Betti numbers; being 1, 0, 22, 0, 1. The definition used in algebraic geometry is that the canonical class is trivial, and H1(X1) = 0.

A Kummer surface is the quotient of a two-dimensional abelian variety A by the action of a → −a. This results in 16 singularities, at the 2-torsion points of A. It was shown classically that such a surface has a birational desingularisation as a quartic in P3, so providing the earliest construction of K3 surfaces. Other constructions are by the intersection of three quadrics (in P5), and as a double cover of the projective plane branched along a sextic curve.

It is known that there is a moduli space for K3 surfaces, of dimension 19. There is a period mapping and Torelli theorem for complex K3 surfaces.

K3 manifolds play an important role in string theory because they provide us with the second simplest compactification after the torus. Compactification on a K3 surface preserves one half of the original supersymmetry.