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Wikipedia / K3 surface

In mathematics, a complex analytic K3 surface is a compact connected complex manifold of dimension 2 with а trivial canonical bundle and irregularity zero. An (algebraic) K3 surface over any field means a smooth, proper, geometrically connected algebraic surface that satisfies the same conditions. In the Enriques–Kodaira classification of surfaces, K3 surfaces form one of the four classes of minimal surfaces of Kodaira dimension zero. A simple example is the Fermat quartic surface in complex projective 3-space.

Together with two-dimensional compact complex tori, K3 surfaces are the Calabi–Yau manifolds (and also the hyperkähler manifolds) of dimension two. As such, they are at the center of the classification of algebraic surfaces, between the positively curved del Pezzo surfaces (which are easy to classify) and the negatively curved surfaces of general type (which are essentially unclassifiable). K3 surfaces can be considered the simplest algebraic varieties whose structure does not reduce to curves or abelian varieties, and yet where a substantial understanding is possible. A complex K3 surface has real dimension 4, and it plays an important role in the study of smooth 4-manifolds. K3 surfaces have been applied to Kac–Moody algebras, mirror symmetry and string theory.

It can be useful to think of complex algebraic K3 surfaces as part of the broader family of complex analytic K3 surfaces. Many other types of algebraic varieties do not have such non-algebraic deformations.

Definition

There are several equivalent ways to define K3 surfaces. The only compact complex surfaces with trivial canonical bundle are K3 surfaces and compact complex tori, and so one can add any condition excluding the latter to define K3 surfaces. For example, it is equivalent to define a complex analytic K3 surface as a simply connected compact complex manifold of dimension 2 with a nowhere-vanishing holomorphic 2-form. (The latter condition says exactly that the canonical bundle is trivial.)

There are also some variants of the definition. Over the complex numbers, some authors consider only the algebraic K3 surfaces. (An algebraic K3 surface is automatically projective.) Or one may allow K3 surfaces to have du Val singularities (the canonical singularities of dimension 2), rather than being smooth.

Calculation of the Betti numbers

The Betti numbers of a complex analytic K3 surface are computed as follows. (A similar argument gives the same answer for the Betti numbers of an algebraic K3 surface over any field, defined using l-adic cohomology is trivial, and the irregularity q(X of the coherent sheaf cohomology) is zero. By Serre duality,

As a result, the arithmetic genus (or holomorphic Euler characteristic) of X is:

On the other hand, the Riemann–Roch theorem (Noether's formula) says:

is the i-th Chern class of the tangent bundle.

Next, the exponential sequence gives an exact sequence is zero, and by Poincaré duality is equal to the topological Euler characteristic

.

Properties

Examples

  • The double cover X of the projective plane branched along a smooth sextic (degree 6) curve is a K3 surface of genus 2 (that is, degree 2g−2 = 2). (This terminology means that the inverse image in X of a general hyperplane is a smooth curve of genus 2.)

  • is a K3 surface of genus 3 (that is, degree 4).

  • A Kummer surface is the quotient of a two-dimensional abelian variety A. This results in 16 singularities, at the 2-torsion points of A. The minimal resolution of this singular surface may also be called a Kummer surface; that resolution is a K3 surface. When A is the Jacobian as a quartic surface with 16 nodes.

  • More generally: for any quartic surface Y with du Val singularities, the minimal resolution of Y is an algebraic K3 surface.

  • The intersection of a quadric is a K3 surface of genus 4 (that is, degree 6).

  • is a K3 surface of genus 5 (that is, degree 8).

  • is a K3 surface, typically of Picard rank 2

  • There are several databases of K3 surfaces with du Val singularities in weighted projective spaces.

The Picard lattice

The Picard group Pic(X) of a complex analytic K3 surface X is the abelian group of complex analytic line bundles on X. For an algebraic K3 surface, Pic(X) is the group of algebraic line bundles on X. The two definitions agree for a complex algebraic K3 surface, by Jean-Pierre Serre's GAGA theorem.

The Picard group of a K3 surface X is always a finitely generated free abelian group; its rank is called the Picard number. In the complex case, Pic(X. It is an important feature of K3 surfaces that many different Picard numbers can occur. For X may also be zero. (In that case, X contains no closed complex curves at all. By contrast, an algebraic surface always contains many continuous families of curves.) Over an algebraically closed field of characteristic p > 0, there is a special class of K3 surfaces, supersingular K3 surfaces, with Picard number 22.

The Picard lattice of a K3 surface is the abelian group Pic(X. Over a general field, the intersection form can be defined using the intersection theory of curves on a surface, by identifying the Picard group with the divisor class group.) The Picard lattice of a K3 surface is always even.

The Hodge index theorem.

(excluding the supersingular case). In particular, algebraic K3 surfaces occur in 19-dimensional families. More details about moduli spaces of K3 surfaces are given below.

The precise description of which lattices can occur as Picard lattices of K3 surfaces is complicated. One clear statement, due to Viacheslav Nikulin and David Morrison is the Picard lattice of some complex projective K3 surface..

Elliptic K3 surfaces

An important subclass of K3 surfaces, easier to analyze than the general case, consists of the K3 surfaces with an elliptic fibration. "Elliptic" means that all but finitely many fibers of this morphism are smooth curves of genus 1. The singular fibers are unions of rational curves (a nodal cubic curve).

Whether a K3 surface is elliptic can be read from its Picard lattice. Namely, in characteristic not 2 or 3, a K3 surface X. (In characteristic 2 or 3, the latter condition may also correspond to a quasi-elliptic fibration.) It follows that having an elliptic fibration is a codimension-1 condition on a K3 surface. So there are 19-dimensional families of complex analytic K3 surfaces with an elliptic fibration, and 18-dimensional moduli spaces of projective K3 surfaces with an elliptic fibration.

Example: Every smooth quartic surface X that contains a line L, given by projecting away from L. The moduli space of all smooth quartic surfaces (up to isomorphism) has dimension 19, while the subspace of quartic surfaces containing a line has dimension 18.

Rational curves on K3 surfaces

In contrast to positively curved varieties such as del Pezzo surfaces, a complex algebraic K3 surface X is not uniruled; that is, it is not covered by a continuous family of rational curves. On the other hand, in contrast to negatively curved varieties such as surfaces of general type, X contains a large discrete set of rational curves (possibly singular). In particular, Fedor Bogomolov and David Mumford showed that every curve on X is linearly equivalent to a positive linear combination of rational curves.

Another contrast to negatively curved varieties is that the Kobayashi metric on a complex analytic K3 surface X is identically zero. The proof uses that an algebraic K3 surface X is always covered by a continuous family of images of elliptic curves. (These curves are singular in X, unless X (where "nondegenerate" means that the derivative of the map is an isomorphism at some point).

The period map

Define a marking of a complex analytic K3 surface X. The space N of marked complex K3 surfaces is a non-Hausdorff complex manifold of dimension 20. The set of isomorphism classes of complex analytic K3 surfaces is the quotient of N by the orthogonal group is far from being properly discontinuous. (For example, the space of smooth quartic surfaces is irreducible of dimension 19, and yet every complex analytic K3 surface in the 20-dimensional family N has arbitrarily small deformations which are isomorphic to smooth quartics.) For the same reason, there is not a meaningful moduli space of compact complex tori of dimension at least 2.

The period mapping sends a K3 surface to its Hodge structure. When stated carefully, the Torelli theorem holds: a K3 surface is determined by its Hodge structure. The period domain is defined as the 20-dimensional complex manifold

sends a marked K3 surface X. This is surjective, and a local isomorphism, but not an isomorphism (in particular because D is Hausdorff and N is not). However, the global Torelli theorem for K3 surfaces says that the quotient map of sets

is bijective. It follows that two complex analytic K3 surfaces X and Y are isomorphic if and only if there is a Hodge isometry.

Moduli spaces of projective K3 surfaces

A polarized K3 surface X of genus g is defined to be a projective K3 surface together with an ample line bundle L such that L. This is also called a polarized K3 surface of degree 2g−2.

Under these assumptions, L is basepoint-free. In characteristic zero, Bertini's theorem implies that there is a smooth curve C in the linear system |L|. All such curves have genus g, which explains why (X,L) is said to have genus g.

The vector space of sections of L has dimension g + 1, and so L gives a morphism from X. In most cases, this morphism is an embedding, so that X is isomorphic to a surface of degree 2g.

There is an irreducible coarse moduli space of polarized complex K3 surfaces of genus g; it can be viewed as a Zariski open subset of a Shimura variety for the group SO(2,19). For each g is a quasi-projective complex variety of dimension 19. Shigeru Mukai showed that this moduli space is unirational. In contrast, Valery Gritsenko, Klaus Hulek is of general type. A survey of this area was given by .

corresponding to K3 surfaces of Picard number at least 2. Those K3 surfaces have polarizations of infinitely many different degrees, not just 2g. However, a concrete version of this idea is the fact that any two complex algebraic K3 surfaces are deformation-equivalent through algebraic K3 surfaces.

More generally, a quasi-polarized K3 surface of genus g means a projective K3 surface with a primitive nef and big line bundle L, but now it may contract finitely many (−2)-curves, so that the image Y of X is singular. (A (−2)-curve with self-intersection −2.) The moduli space of quasi-polarized K3 surfaces of genus g is still irreducible of dimension 19 (containing the previous moduli space as an open subset). Formally, it works better to view this as a moduli space of K3 surfaces Y with du Val singularities.

The ample cone and the cone of curves

A remarkable feature of algebraic K3 surfaces is that the Picard lattice determines many geometric properties of the surface, including the convex cone with positive self-intersection has two connected componentss. Call the positive cone the component that contains any ample divisor on X.

Case 1: There is no element u of Pic(X. Then the ample cone is equal to the positive cone. Thus it is the standard round cone.

, the set of roots of the Picard lattice. The orthogonal complements of the roots form a set of hyperplanes which all go through the positive cone. Then the ample cone is a connected component of the complement of these hyperplanes in the positive cone. Any two such components are isomorphic via the orthogonal group of the lattice Pic(X), since that contains the reflection across each root hyperplane. In this sense, the Picard lattice determines the ample cone up to isomorphism.

A related statement, due to Sándor Kovács, is that knowing one ample divisor A in Pic(X) determines the whole cone of curves of X. Namely, suppose that X. In the first case, X contains no (−2)-curves; in the second case, the closed cone of curves is the closed convex cone spanned by all (−2)-curves., there is one other possibility: the cone of curves may be spanned by one (−2)-curve and one curve with self-intersection 0.) So the cone of curves is either the standard round cone, or else it has "sharp corners" (because every (−2)-curve spans an isolated extremal ray of the cone of curves).

Automorphism group

K3 surfaces are somewhat unusual among algebraic varieties in that their automorphism groups may be infinite, discrete, and highly nonabelian. By a version of the Torelli theorem, the Picard lattice of a complex algebraic K3 surface X determines the automorphism group of X up to commensurability. Namely, let the Weyl group W be the subgroup of the orthogonal group O(Pic(X. Then W is a normal subgroup of O(Pic(X)), and the automorphism group of X is commensurable with the quotient group O(Pic(X))/W. A related statement, due to Hans Sterk, is that Aut(X) acts on the nef cone of X with a rational polyhedral fundamental domain.

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 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 string compactified on a K3 surface is equivalent to the heterotic string compactified on a 4-torus ().

History

were studied by Ernst Kummer, Arthur Cayley, Friedrich Schur and other 19th-century geometers. More generally, Federigo Enriques observed in 1893 that for various numbers g, there are surfaces of degree 2g with trivial canonical bundle and irregularity zero., and Francesco Severi showed that the moduli space of such surfaces has dimension 19 for each g.

André gave K3 surfaces their name (see the quotation above) and made several influential conjectures about their classification. Kunihiko Kodaira completed the basic theory around 1960, in particular making the first systematic study of complex analytic K3 surfaces which are not algebraic. He showed that any two complex analytic K3 surfaces are deformation-equivalent and hence diffeomorphic, which was new even for algebraic K3 surfaces. An important later advance was the proof of the Torelli theorem for complex algebraic K3 surfaces by Ilya Piatetski-Shapiro and Igor Shafarevich (1971), extended to complex analytic K3 surfaces by Daniel Burns and Michael Rapoport (1975).

See also

Notes

References