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 5 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, and the mountain K2 in Kashmir.

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.

Properties

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

Siu (1983) showed that all complex K3 surfaces are Kähler manifolds. As a consequence of this and Yau's solution to the Calabi conjecture, they all admit Ricci-flat metrics.

The period map

There is a coarse moduli space for marked complex K3 surfaces, a non-Hausdorff smooth analytic space of dimension 20. There is a period mapping and Torelli theorem 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 KΩ⁰ 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
• KΩ⁰ is the set of elements (κ, [ω]) of KΩ such that (κd) ≠ 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 surfaces, then the curves in the linear system have genus g where c₁²(L) =2g-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 c₁²(L) =2g-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.

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 P³ is a genus 3 K3 surface.
• The intersection of a quadric and a cubic in P⁴ gives genus 4 K3 surfaces.
• The intersection of three quadrics in P⁵ gives genus 5 K3 surfaces.
• Brown (2007) describes a computer database of K3 surfaces.

See also

• Supersingular K3 surface
• Classification of algebraic surfaces

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, MR728990, http://www.numdam.org/item?id=SB_1982-1983__25__217_0 
• Beauville, A.; Bourguignon, J.-P.; Demazure, M. (1985), Géométrie des surfaces K3: modules et périodes, Séminaires Palaiseau, Astérisque, 126, Paris: Société Mathématique de France, MR785216 
• Brown, Gavin (2007), "A database of polarized K3 surfaces", Experimental Mathematics 16 (1): 7–20, doi:10.1080/10586458.2007.10128983, MR2312974, http://projecteuclid.org/euclid.em/1175789798 
• 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, MR0447635, http://www.numdam.org/item?id=ASENS_1975_4_8_2_235_0 
• 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, MR2306149 
• 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, MR2336040 
• 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, MR2310254 
• Rudakov, A.N. (2001), "K3 surface", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1556080104, http://www.encyclopediaofmath.org/index.php?title=k/k055040 
• 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, MR0284440 
• Siu, Y. T. (1983), "Every K3 surface is Kähler", Inventiones Mathematicae 73 (1): 139–150, Bibcode 1983InMat..73..139S, doi:10.1007/BF01393829, MR707352 
• 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.)", Astérisque, Séminaire Bourbaki. 2006/2007. Exp 981 (317): 467–490, ISBN 978-2-85629-253-2, MR2487743, http://www.bourbaki.ens.fr/TEXTES/981.pdf 
• 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, MR537935 

External links

• graded ring database homepage with a catalog of K3 surfaces
• K3 Surfaces and String Duality, by Paul Aspinwall
• The Geometry of K3 surfaces, by David Morrison
• K3 database for the Magma computer algebra system