Definitions

K3 surface

In mathematics, in the field of complex manifolds, a K3 surface is an important and interesting example of a compact complex surface (complex dimension 2 being real dimension 4).

Together with two-dimensional complex tori, they are the Calabi-Yau manifolds of dimension two. Most K3 surfaces, in a definite sense, are not algebraic. This means that, in general, they cannot be embedded in any projective space as a surface defined by polynomial equations. However, K3 surfaces first arose in algebraic geometry and it is in this context that they received their name — it is after three algebraic geometers, Kummer, Kähler and Kodaira, and alludes to the mountain peak K2, which was in the news when the name was given during the 1950s.

Definition

There are many equivalent properties that can be used to characterize a K3 surface. The definition given depends on the context:

In differential geometry, a typical definition is of "a compact, complex, simply connected surface with trivial canonical bundle".

In algebraic geometry, the definition "a surface, X, with trivial canonical class such that H1(X,OX) = 0." is preferred since it generalizes to more arbitrary base fields (not just the complex numbers). Here, H1(X,OX) denotes the first sheaf cohomology group of OX, the sheaf of regular functions on X.

Another characterization, sometimes found in physics literature, is that a K3 surface is a Calabi-Yau manifold of two complex dimensions that is not T4.

Important properties

All K3 surfaces are diffeomorphic to one another and so have the same Betti numbers: 1, 0, 22, 0, 1.

All K3 surfaces are Kähler manifolds.

As a consequence of Yau's solution to the Calabi conjecture, all K3 surfaces admit Ricci-flat metrics.

It is known that there is a coarse moduli space for complex K3 surfaces, of dimension 20. There is a period mapping and Torelli theorem for complex K3 surfaces. There are also several other types of moduli for K3 surfaces which admit good period maps.

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.

Examples

1. A Kummer surface is the quotient of a two-dimensional abelian variety A by the action a → −a. This results in 16 singularities, at the 2-torsion points of A. It was shown classically that the minimal resolution of this quotient is a K3 surface.
2. A non-singular degree 4 surface in P3.
3. The intersection of a quadric and a cubic in P4.
4. The intersection of three quadrics in P5.
5. A double cover of the projective plane branched along a non-singular degree 6 curve.