- K3 surface
In
mathematics , in the field ofcomplex manifold s, a K3 surface is an important and interesting example of a compact complex surface (complex dimension 2 beingreal dimension 4).Together with two-dimensional
complex tori , they are theCalabi-Yau manifold s 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 inalgebraic 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 peakK2 , 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 trivialcanonical bundle ".In
algebraic geometry , the definition "a surface, "X", with trivialcanonical class such that "H"1("X","O"X) = 0." is preferred since it generalizes to more arbitrary base fields (not just thecomplex numbers ). Here, "H"1("X","O"X) denotes the firstsheaf 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 "T4".Important properties
All K3 surfaces are diffeomorphic to one another and so have the same
Betti number s: 1, 0, 22, 0, 1.All K3 surfaces are
Kähler manifold s.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 aperiod mapping andTorelli 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 thetorus . Compactification on a K3 surface preserves one half of the originalsupersymmetry .Examples
#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.
#Anon-singular degree 4 surface in "P"3.
#The intersection of aquadric and a cubic in "P"4.
#The intersection of threequadric s in "P"5.
#Adouble cover of theprojective plane branched along anon-singular degree 6 curve.ee also
*
Supersingular K3 surface
*Classification of algebraic surfaces References
*citation|title=Compact Complex Surfaces|first= Wolf P.|last1= Barth|first2=Klaus|last2= Hulek|first3= Chris A.M. |last3=Peters|first4= Antonius |last4=Van de Ven |ISBN=3-540-00832-2|year=2004
*springer|title=K3 surface|id=k/k055040|first=A.N. |last=RudakovExternal links
* [http://arxiv.org/abs/hep-th/9611137 K3 Surfaces and String Duality, by Paul Aspinwall]
* [http://www.cgtp.duke.edu/ITP99/morrison/cortona.pdf The Geometry of K3 surfaces, by David Morrison]
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