nLab
K3 surface

Contents

Contents

Definition

A K3 surface is a Calabi-Yau variety of dimension 22 (a Calabi-Yau algebraic surface/complex surface). This means that the canonical bundle ω X= 2Ω X𝒪 X\omega_X=\wedge^2\Omega_X\simeq \mathcal{O}_X is trivial and H 1(X,𝒪 X)=0H^1(X, \mathcal{O}_X)=0.

The term “K3” is

in honor of Kummer, Kähler, Kodaira, and the beautiful K2 mountain in Kashmir

(Weil 79, p. 546)

Examples

  • A cyclic cover of 2\mathbb{P}^2 branched over a curve of degree 66.

  • A nonsingular degree 44 hypersurface in 3\mathbb{P}^3, such as the Fermat quartic? {[w,x,y,z] 3|w 4+x 4+y 4+z 4=0}\{[w,x,y,z] \in \mathbb{P}^3 | w^4 + x^4 + y^4 + z^4 = 0\} (in fact every K3 surface over \mathbb{C} is diffeomorphic to this example).

  • The flat orbifold quotient of the 4-torus by the sign involution on all four canonical coordinates is the flat compact 4-dimensional orbifold known as the Kummer surface T 4 2T^4 \sslash \mathbb{Z}_2, a singular K3-surface (e.g. Bettiol-Derdzinski-Piccione 18, 5.5)

Properties

Basic properties

Cohomology

Proposition

(integral cohomology of K3-surface)

The integral cohomology of a K3-surface XX is

H n(X,){ | n=0 0 | n=1 22 | n=2 0 | n=3 | n=4 H^n(X,\mathbb{Z}) \;\simeq\; \left\{ \array{ \mathbb{Z} &\vert& n = 0 \\ 0 &\vert& n = 1 \\ \mathbb{Z}^{22} &\vert& n = 2 \\ 0 &\vert& n = 3 \\ \mathbb{Z} &\vert& n = 4 } \right.

(e.g. Barth-Peters-Van den Ven 84, VIII Prop. 3.2)

Proposition

(Betti numbers of a K3-surface)

The Hodge diamond? is completely determined (even in positive characteristic) and hence the Hodge-de Rham spectral sequence degenerates at E 1E_1. This also implies that the Betti numbers are completely determined as 1,0,22,0,11, 0, 22, 0, 1:

h 0,0 h 1,0 h 0,1 h 2,0 h 1,1 h 0,2 h 2,1 h 1,2 h 2,2= 1 0 0 1 20 1 0 0 1 \array{ && h^{0,0} \\ & h^{1,0} && h^{0,1} \\ h^{2,0} & & h^{1,1} & & h^{0,2} \\ & h^{2,1} & & h^{1,2} \\ && h^{2,2} } \;\;\;=\;\;\; \array{ && 1 \\ & 0 && 0 \\ 1 & & 20 & & 1 \\ & 0 & & 0 \\ && 1 }

(e.g. Barth-Peters-Van den Ven 84, VIII Prop. 3.3)

Moduli of higher line bundles and deformation theory

In positive characteristic pp:

The Néron-Severi group of a K3 is a free abelian group

The formal Brauer group is

  • either the formal additive group, in which case it has height h=h = \infty, by definition;

  • or its height is 1h101 \leq h \leq 10, and every value may occur

(Artin 74), see also (Artin-Mazur 77, p. 5 (of 46))

moduli spaces of line n-bundles with connection on nn-dimensional XX

nnCalabi-Yau n-foldline n-bundlemoduli of line n-bundlesmoduli of flat/degree-0 n-bundlesArtin-Mazur formal group of deformation moduli of line n-bundlescomplex oriented cohomology theorymodular functor/self-dual higher gauge theory of higher dimensional Chern-Simons theory
n=0n = 0unit in structure sheafmultiplicative group/group of unitsformal multiplicative groupcomplex K-theory
n=1n = 1elliptic curveline bundlePicard group/Picard schemeJacobianformal Picard groupelliptic cohomology3d Chern-Simons theory/WZW model
n=2n = 2K3 surfaceline 2-bundleBrauer groupintermediate Jacobianformal Brauer groupK3 cohomology
n=3n = 3Calabi-Yau 3-foldline 3-bundleintermediate JacobianCY3 cohomology7d Chern-Simons theory/M5-brane
nnintermediate Jacobian

References

General

Original sources include

  • Michael Artin, Supersingular K3 Surfaces, Annal. Sc. d, l’Éc Norm. Sup. 4e séries, T. 7, fasc. 4, 1974, pp. 543-568

  • Andre Weil, Final report on contract AF 18 (603)-57. In Scientific works. Collected papers. Vol. II (1951-1964). 1979.

Textbook accounts include

  • W. Barth, C. Peters, A. Van den Ven, chapter VII of Compact complex surfaces, Springer 1984

Lecture notes include

Discussion of the deformation theory of K3-surfaces (of their Picard schemes) is (see also at Artin-Mazur formal group) in

In string theory

In string theory, the KK-compactification of type IIA string theory/M-theory/F-theory on K3-fibers is supposed to exhibit te duality between M/F-theory and heterotic string theory, originally due to

Review includes

Further discussion includes

Specifically in relation to the putative K-theory-classification of D-brane charge:

Specifically in M-theory on G2-manifolds:

Specifically in relation to Moonshine:

Specifically in relation to little string theory:

Last revised on January 24, 2019 at 03:56:26. See the history of this page for a list of all contributions to it.