In the second part of my report, we deal with the Kähler varieties known as K3, named in honor of Kummer, Kähler, Kodaira and of the beautiful mountain K2 in Kashmir.
André Weil (1958, p. 546), describing the reason for the name "K3 surface"
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.
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
and, from Serre duality,
Altogether, one obtains the Euler characteristic
On the other hand, the Riemann–Roch theorem (Noether's formula) reads
where ci(S) is the i-th Chern class. Since KS is trivial, its first Chern class c1(S) vanishes, hence c2(S) = 24.
Since c2(S) is equal to the Euler number e(S) = b0(S) − b1(S) + b2(S) − b3(S) + b4(S) and b0(S) = b4(S) = 1, b1(S) = b3(S) = 2q(S) = 0, we obtain b2(S) = 22.
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 groups are indicated in the Hodge diamond
3. On , the cup product 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.
The set M of pairs consisting of a complex K3 surface S and a Kähler class of H1,1(S,R) 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 II3,19.
- 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 surface, 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 Fg 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 Fg 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 E8×E8 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).
- 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.
K3 surface was first discovered by Ramanujan in 1910s but remained unpublished, and was rediscovered by André Weil (1958) who named them in honor of three algebraic geometers, Kummer, Kähler and Kodaira, and the mountain K2 in Pakistan.
- Supersingular K3 surface
- Classification of algebraic surfaces
- Mathieu moonshine, a mysterious relationship between K3 surfaces and the Mathieu group M24.
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