K3 surface
Encyclopedia
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 manifold
s 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 named them in honor of three algebraic geometers, Kummer
, Kähler
and Kodaira, and the mountain K2
in Kashmir.
There are a few variations of the definition: some authors restrict to projective surfaces, and some allow surfaces with
Du Val singularities.
to one another and so have the same Betti number
s: 1, 0, 22, 0, 1.
The Hodge diamond is
showed that all complex K3 surfaces are Kähler manifold
s. As a consequence of this and Yau's solution to the Calabi conjecture
, they all admit Ricci-flat metrics.
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 H1,1(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:
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. showed that this moduli space Fg is unirational if g≤13, and showed that it is of general type if g≥63. gave a survey of this area.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a K3 surface is a complex or algebraic smooth minimal complete surface that is regular
Irregularity of a surface
In mathematics, the irregularity of a complex surface X is the Hodge number h0,1= dim H1, usually denoted by q . The irregularity of an algebraic surface is sometimes defined to be this Hodge number, and sometimes defined to be the dimension of the Picard variety , which is the same in...
and has trivial canonical bundle
Canonical bundle
In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n is the line bundle\,\!\Omega^n = \omegawhich is the nth exterior power of the cotangent bundle Ω on V. Over the complex numbers, it is the determinant bundle of holomorphic n-forms on V.This is the dualising...
.
In the Enriques-Kodaira classification of surfaces they form one of the 5 classes of surfaces of Kodaira dimension
Kodaira dimension
In algebraic geometry, the Kodaira dimension κ measures the size of the canonical model of a projective variety V.The definition of Kodaira dimension, named for Kunihiko Kodaira, and the notation κ were introduced in the seminar.-The plurigenera:...
0.
Together with two-dimensional complex tori, they are the Calabi-Yau manifold
Calabi-Yau manifold
A Calabi-Yau manifold is a special type of manifold that shows up in certain branches of mathematics such as algebraic geometry, as well as in theoretical physics...
s 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 named them in honor of three algebraic geometers, Kummer
Ernst Kummer
Ernst Eduard Kummer was a German mathematician. Skilled in applied mathematics, Kummer trained German army officers in ballistics; afterwards, he taught for 10 years in a gymnasium, the German equivalent of high school, where he inspired the mathematical career of Leopold Kronecker.-Life:Kummer...
, Kähler
Erich Kähler
was a German mathematician with wide-ranging geometrical interests.Kähler was born in Leipzig, and studied there. He received his Ph.D. in 1928 from the University of Leipzig. He held professorial positions in Königsberg, Leipzig, Berlin and Hamburg...
and Kodaira, and the mountain K2
K2
K2 is the second-highest mountain on Earth, after Mount Everest...
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 diffeomorphicDiffeomorphism
In mathematics, a diffeomorphism is an isomorphism in the category of smooth manifolds. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth.- Definition :...
to one another and so have the same Betti number
Betti number
In algebraic topology, a mathematical discipline, the Betti numbers can be used to distinguish topological spaces. Intuitively, the first Betti number of a space counts the maximum number of cuts that can be made without dividing the space into two pieces....
s: 1, 0, 22, 0, 1.
The Hodge diamond is
| 1 | ||||
|---|---|---|---|---|
| 0 | 0 | |||
| 1 | 20 | 1 | ||
| 0 | 0 | |||
| 1 |
showed that all complex K3 surfaces are Kähler manifold
Kähler manifold
In mathematics, a Kähler manifold is a manifold with unitary structure satisfying an integrability condition.In particular, it is a Riemannian manifold, a complex manifold, and a symplectic manifold, with these three structures all mutually compatible.This threefold structure corresponds to the...
s. As a consequence of this and Yau's solution to the Calabi conjecture
Calabi conjecture
In mathematics, the Calabi conjecture was a conjecture about the existence of good Riemannian metrics on complex manifolds, made by and proved by ....
, 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 mappingPeriod mapping
In mathematics, in the field of algebraic geometry, the period mapping relates families of Kähler manifolds to families of Hodge structures.- Ehresmann's theorem :...
and Torelli theorem
Torelli theorem
In mathematics, the Torelli theorem is a classical result of algebraic geometry over the complex number field, stating that a non-singular projective algebraic curve C is determined by its Jacobian variety J, when the latter is given in the form of a principally polarized abelian variety...
for complex K3 surfaces.
If M is the set of pairs consisting of a complex K3 surface S and a Kähler class of H1,1(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 latticeUnimodular latticeIn mathematics, a unimodular lattice is a lattice of determinant 1 or −1.The E8 lattice and the Leech lattice are two famous examples.- Definitions :...
II3,19 - Ω is the Hermitian symmetric spaceHermitian symmetric spaceIn mathematics, a Hermitian symmetric space is a Kähler manifold M which, as a Riemannian manifold, is a Riemannian symmetric space. Equivalently, M is a Riemannian symmetric space with a parallel complex structure with respect to which the Riemannian metric is Hermitian...
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 wherec12(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. showed that this moduli space Fg is unirational if g≤13, and showed that it is of general type if g≥63. gave a survey of this area.
Examples
- A double cover of the projective planeProjective planeIn mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines that do not intersect...
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 varietyAbelian varietyIn mathematics, particularly in algebraic geometry, complex analysis and number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions...
A by the action a → −a. This results in 16 singularities, at the 2-torsion points of A. The minimal resolutionResolution of singularitiesIn algebraic geometry, the problem of resolution of singularities asks whether every algebraic variety V has a resolution, a non-singular variety W with a proper birational map W→V...
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 quadricQuadricIn mathematics, a quadric, or quadric surface, is any D-dimensional hypersurface in -dimensional space defined as the locus of zeros of a quadratic polynomial...
and a cubic in P4 gives genus 4 K3 surfaces. - The intersection of three quadricQuadricIn mathematics, a quadric, or quadric surface, is any D-dimensional hypersurface in -dimensional space defined as the locus of zeros of a quadratic polynomial...
s in P5 gives genus 5 K3 surfaces. describes a computer database of K3 surfaces.
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 systemMagma computer algebra systemMagma is a computer algebra system designed to solve problems in algebra, number theory, geometry and combinatorics. It is named after the algebraic structure magma...