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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). ## Definition

## Important properties

## Examples

## See also

## References

## External links

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.

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 H^{1}(X,O_{X}) = 0." is preferred since it generalizes to more arbitrary base fields (not just the complex numbers). Here, H^{1}(X,O_{X}) denotes the first sheaf cohomology group of O_{X}, 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 T^{4}.

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.

- 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.
- A non-singular degree 4 surface in P
^{3}. - The intersection of a quadric and a cubic in P
^{4}. - The intersection of three quadrics in P
^{5}. - A double cover of the projective plane branched along a non-singular degree 6 curve.

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Last updated on Saturday September 13, 2008 at 07:10:06 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Saturday September 13, 2008 at 07:10:06 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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