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# K3 surface

In mathematics, a K3 surface is a complex or algebraic smooth minimal complete surface that is regular and has trivial canonical bundle.

In the Enriques–Kodaira classification of surfaces they form one of the 4 classes of surfaces of Kodaira dimension 0.

Together with two-dimensional complex tori, they are the Calabi–Yau manifolds of dimension two. Most complex K3 surfaces are not algebraic. This means that they cannot be embedded in any projective space as a surface defined by polynomial equations. André Weil (1958) named them in honor of three algebraic geometers, Kummer, Kähler and Kodaira

Definition

There are many equivalent properties that can be used to characterize a K3 surface. The only complete smooth surfaces with trivial canonical bundle are K3 surfaces and tori (or abelian varieties), so one can add any condition to exclude the latter to define K3 surfaces. Over the complex numbers the condition that the surface is simply connected is sometimes used.

There are a few variations of the definition: some authors restrict to projective surfaces, and some allow surfaces with Du Val singularities.

Calculation of the Betti numbers

Equivalently to the above definition, K3 surface S is defined as the surface that has trivial canonical bundle KS = 0 and irregularity q = 0. Then there is a trivial mapping from S to P1 and \( q = h^{0,1} = \text{dim} H^1(S,\mathcal{O}_S) =0 \).

From Serre duality

\( h^2(S,\mathcal{O}_S)=h^0(S,K_S)=1. \)

Combining with this, one obtains the Euler characteristic

\( \chi(S,\mathcal{O}_S):=\sum_i (-1)^i h^i(S,\mathcal{O}_S)=2 \).

On the other hand, from the Riemann-Roch Theorem (Noether's formula)

\( \chi(S,\mathcal{O}_S)=\frac{1}{12}(c_1(S)^2+c_2(S)) \)

where ci denotes i-th Chern class. KS is trivial and so the first Chern class c1 = 0. The Euler number e(S) is equal to the second Chern class c2(S) and then e(S) = 24. Therefore, b1 = 0, b2 = 22.

Properties

1. All complex K3 surfaces are diffeomorphic to one another (proved by Kunihiko Kodaira firstly). Siu (1983) showed that all complex K3 surfaces are Kähler manifolds. As a consequence of this and Yau's solution to the Calabi conjecture, all complex K3 surfaces admit Ricci-flat metrics.

2. The (p,q)-th cohomology group is well known. The Hodge diamond is

1 | ||||

0 | 0 | |||

1 | 20 | 1 | ||

0 | 0 | |||

1 |

3. On \( H^2(S,\mathbb{Z}) \) this defines a lattice structure, called the K3 lattice, as described in the next section.

Because of the above properties, K3 surfaces have been studied extensively not only in algebraic geometry but also in Kac–Moody algebras, mirror symmetry and string theory. In particular, the lattice structure provides the modularity with the Néron–Severi group on it.

The period map

There is a coarse moduli space for marked complex K3 surfaces, a non-Hausdorff smooth analytic space of complex dimension 20. There is a period mapping and the Torelli theorem holds for complex K3 surfaces.

If *M* is the set of pairs consisting of a complex K3 surface *S* and a Kähler class of H^{1,1}(*S*,**R**) then *M* is in a natural way a real analytic manifold of dimension 60. There is a refined period map from *M* to a space KΩ^{0} that is an isomorphism. The space of periods can be described explicitly as follows:

*L*is the even unimodular lattice II_{3,19}.- Ω is the Hermitian symmetric space consisting of the elements of the complex projective space of
*L*⊗**C**that are represented by elements ω with (ω,ω)=0, (ω,ω^*)>0. - KΩ is the set of pairs (κ, [ω]) in (
*L*⊗**R**, Ω) with (κ,E(ω))=0, (κ,κ)>0. - KΩ
^{0}is the set of elements (κ, [ω]) of KΩ such that (κ*d*) ≠ 0 for every*d*in*L*with (*d*,*d*)=−2, (ω,*d*)=0.

Projective K3 surfaces

If *L* is a line bundle on a K3 surface, then the curves in the linear system have genus *g* where *c*_{1}^{2}(*L*) =2*g*-2. A K3 surface with a line bundle *L* like this is called a K3 surface of genus *g*. A K3 surface may have many different line bundles making it into a K3 surface of genus *g* for many different values of *g*. The space of sections of the line bundle has dimension *g*+1, so there is a morphism of the K3 surface to projective space of dimension *g*. There is a moduli space *F*_{g} of K3 surfaces with a primitive ample line bundle *L* with *c*_{1}^{2}(*L*) =2*g*-2, which is nonempty of dimension 19 for *g*≥ 2. Mukai (2006) showed that this moduli space *F*_{g} is unirational if *g*≤13, and V. A. Gritsenko, Klaus Hulek, and G. K. Sankaran (2007) showed that it is of general type if *g*≥63. Voisin (2008) gave a survey of this area.

Relation to string duality

K3 surfaces appear almost ubiquitously in string duality and provide an important tool for the understanding of it. String compactifications on these surfaces are not trivial, yet they are simple enough for us to analyze most of their properties in detail. The type IIA string, the type IIB string, the E_{8}×E_{8} heterotic string, the Spin(32)/Z2 heterotic string, and M-theory are related by compactification on a K3 surface. For example, the Type IIA compactified on a K3 surface is equivalent to the heterotic string compactified on 4-torus Aspinwall (1996).

Examples

- A double cover of the projective plane branched along a non-singular degree 6 curve is a genus 2 K3 surface.
- 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*. The minimal resolution of this quotient is a genus 3 K3 surface. - A non-singular degree 4 surface in
**P**^{3}is a genus 3 K3 surface. - The intersection of a quadric and a cubic in
**P**^{4}gives genus 4 K3 surfaces. - The intersection of three quadrics in
**P**^{5}gives genus 5 K3 surfaces. - Brown (2007) describes a computer database of K3 surfaces.

See also

Supersingular K3 surface

Classification of algebraic surfaces

Umbral moonshine, a mysterious relationship between K3 surfaces and the Mathieu group M24.

References

Barth, Wolf P.; Hulek, Klaus; Peters, Chris A.M.; Van de Ven, Antonius (2004), Compact Complex Surfaces, Berlin: Springer, ISBN 3-540-00832-2

Beauville, Arnaud (1983), "Surfaces K3", Bourbaki seminar, Vol. 1982/83 Exp 609, Astérisque 105, Paris: Société Mathématique de France, pp. 217–229, MR 728990

Beauville, A.; Bourguignon, J.-P.; Demazure, M. (1985), Géométrie des surfaces K3: modules et périodes, Séminaires Palaiseau, Astérisque 126, Paris: Société Mathématique de France, MR 785216

Brown, Gavin (2007), "A database of polarized K3 surfaces", Experimental Mathematics 16 (1): 7–20, doi:10.1080/10586458.2007.10128983, MR 2312974

Burns, Dan; Rapoport, Michael (1975), "On the Torelli problem for kählerian K-3 surfaces", Annales Scientifiques de l'École Normale Supérieure. Quatrième Série 8 (2): 235–273, MR 0447635

Dolgachev, Igor V.; Kondo, Shigeyuki (2007), "Moduli of K3 surfaces and complex ball quotients", in Rolf-Peter Holzapfel, A. Muhammed Uludağ and Masaaki Yoshida, Arithmetic and geometry around hypergeometric functions, Progr. Math. 260, Basel, Boston, Berlin: Birkhäuser, pp. 43–100, arXiv:math/0511051, Bibcode:2005math.....11051D, ISBN 978-3-7643-8283-4, MR 2306149

Gritsenko, V. A.; Hulek, Klaus; Sankaran, G. K. (2007), "The Kodaira dimension of the moduli of K3 surfaces", Inventiones Mathematicae 169 (3): 519–567, arXiv:math/0607339, Bibcode:2007InMat.169..519G, doi:10.1007/s00222-007-0054-1, MR 2336040

Mukai, Shigeru (2006), "Polarized K3 surfaces of genus thirteen", Moduli spaces and arithmetic geometry, Adv. Stud. Pure Math. 45, Tokyo: Math. Soc. Japan, pp. 315–326, MR 2310254

Rudakov, A.N. (2001), "K3 surface", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Pjateckiĭ-Šapiro, I. I.; Šafarevič, I. R. (1971), "Torelli's theorem for algebraic surfaces of type K3", Math Ussr Izv, 5 (3): 547–588, Bibcode:1971IzMat...5..547P, doi:10.1070/IM1971v005n03ABEH001075, MR 0284440

Siu, Y. T. (1983), "Every K3 surface is Kähler", Inventiones Mathematicae 73 (1): 139–150, Bibcode:1983InMat..73..139S, doi:10.1007/BF01393829, MR 707352

Voisin, Claire (2008), "Géométrie des espaces de modules de courbes et de surfaces K3 (d'après Gritsenko-Hulek-Sankaran, Farkas-Popa, Mukai, Verra, et al.)" (PDF), Astérisque, Séminaire Bourbaki. 2006/2007. Exp 981 (317): 467–490, ISBN 978-2-85629-253-2, MR 2487743

Weil, André (1958), "Final report on contract AF 18(603)-57", Scientific works. Collected papers II, Berlin, New York: Springer-Verlag, pp. 390–395, 545–547, ISBN 978-0-387-90330-9, MR 537935

Aspinwall, Paul (1996). "K3 Surfaces and String Duality". arXiv:hep-th/9611137.

External links

Graded Ring Database homepage for a catalog of K3 surfaces

The Geometry of K3 surfaces, by David Morrison

K3 database for the Magma computer algebra system

Undergraduate Texts in Mathematics

Graduate Studies in Mathematics

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