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.Huybrechts (2016), Remark 1.1.2) 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.Huybrechts (2016), section 2.3. (A similar argument gives the same answer for the Betti numbers of an algebraic K3 surface over any field, defined using l-adic cohomology.) By definition, the canonical bundle is trivial, and the irregularity q(X) (the dimension of the coherent sheaf cohomology group ) 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:
where is the i-th Chern class of the tangent bundle. Since is trivial, its first Chern class is zero, and so .
Next, the exponential sequence gives an exact sequence of cohomology groups , and so . Thus the Betti number is zero, and by Poincaré duality, is also zero. Finally, is equal to the topological Euler characteristic
Since and , it follows that .Huybrechts (2016), section 2.4.
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 in is a smooth curve of genus 2.) A smooth quartic (degree 4) surface in is a K3 surface of genus 3 (that is, degree 4). A Kummer surface is the quotient of a two-dimensional abelian variety A by the action . 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 of a curve of genus 2, Kummer showed that the quotient can be embedded into 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 and a cubic in is a K3 surface of genus 4 (that is, degree 6). The intersection of three quadrics in is a K3 surface of genus 5 (that is, degree 8). The complete intersection of two bihomogeneous forms of bidegrees and in 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) is a subgroup of . It is an important feature of K3 surfaces that many different Picard numbers can occur. For X a complex algebraic K3 surface, can be any integer between 1 and 20. In the complex analytic case, 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) together with its intersection form, a symmetric bilinear form with values in the integers. (Over , the intersection form is the restriction of the intersection form on . 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, meaning that the integer is even for each .
The Hodge index theorem implies that the Picard lattice of an algebraic K3 surface has signature . Many properties of a K3 surface are determined by its Picard lattice, as a symmetric bilinear form over the integers. This leads to a strong connection between the theory of K3 surfaces and the arithmetic of symmetric bilinear forms. As a first example of this connection: a complex analytic K3 surface is algebraic if and only if there is an element with .Barth et al. (2004), Theorem 6.1.
Roughly speaking, the space of all complex analytic K3 surfaces has complex dimension 20, while the space of K3 surfaces with Picard number has dimension (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 that every even lattice of signature with is the Picard lattice of some complex projective K3 surface.Huybrechts (2016), Corollary 14.3.1 and Remark 14.3.7. The space of such surfaces has dimension .
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, with the possible types of singular fibers classified by Kodaira. There are always some singular fibers, since the sum of the topological Euler characteristics of the singular fibers is . A general elliptic K3 surface has exactly 24 singular fibers, each of type (a nodal cubic curve).Huybrechts (2016), Remark 11.1.12.
Whether a K3 surface is elliptic can be read from its Picard lattice. Namely, in characteristic not 2 or 3, a K3 surface X has an elliptic fibration if and only if there is a nonzero element with .Huybrechts (2016), Proposition 11.1.3. (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 in that contains a line L has an elliptic fibration , 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.Huybrechts (2016), Corollary 13.1.5.
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.Kamenova et al. (2014), Corollary 2.2; Huybrechts (2016), Corollary 13.2.2. (These curves are singular in X, unless X happens to be an elliptic K3 surface.) A stronger question that remains open is whether every complex K3 surface admits a nondegenerate holomorphic map from (where "nondegenerate" means that the derivative of the map is an isomorphism at some point).Huybrechts (2016), section 13.0.3.
The period map Define a marking of a complex analytic K3 surface X to be an isomorphism of lattices from to the K3 lattice . The space N of marked complex K3 surfaces is a non-Hausdorff complex manifold of dimension 20.Huybrechts (2016), section 6.3.3. The set of isomorphism classes of complex analytic K3 surfaces is the quotient of N by the orthogonal group , but this quotient is not a geometrically meaningful moduli space, because the action of is far from being properly discontinuous.Huybrechts (2016), section 6.3.1 and Remark 6.3.6. (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.Huybrechts (2016), section 7.1.3.) 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
The period mapping sends a marked K3 surface X to the complex line . 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 from to , that is, an isomorphism of abelian groups that preserves the intersection form and sends to .Huybrechts (2016), Theorem 7.5.3.
Moduli spaces of projective K3 surfaces