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)


Characteristic classes

We discuss some characteristic classes of (the tangent bundle of) K3, and their evaluation on (the fundamental class of) K3K3 (i.e. their integration over K3K3).

Of K3K3

Proposition

(Euler characteristic of K3)

The Euler characteristic of K3 is 24:

χ 4[K3]=24. \chi_4[K3] \;=\; 24 \,.
Proposition

(first Chern class of K3)

The first Chern class of K3 vanishes:

c 1(K3)=0. c_1(K3) \;=\; 0 \,.
Proposition

(second Chern class of K3)

The second Chern class of K3 evaluates to 24:

c 2[K3]=24. c_2[K3] \;=\; 24 \,.
Proof

For every closed complex manifold the evaluation of the top degree Chern class equals the Euler characteristic. Hence the statement follows from Prop. .

Proposition

(first Chern class of K3)

The first Pontryagin class of K3 evaluates to 48:

p 1[K3]=48. p_1[K3] \;=\; -48 \,.

(see also e.g. Duff-Liu-Minasian 95 (5.10))

Proof

For a complex manifold the first Pontryagin class is the following polynomial in the first Chern class and the second Chern class. By Prop. the first Chern class vanishes, and by Prop. the second Chern class evaluates to 24:

p 1[K3] =c 1c 1=0[K3]2c 2[K3]24 =48. \begin{aligned} p_1[K3] & =\; \underset{= 0}{\underbrace{c_1 \cup c_1}}[K3] - 2 \underset{24}{\underbrace{c_2[K3]}} \\ & = - 48 \end{aligned} \,.

Of K3×K3K3 \times K3

Now consider the Cartesian product space K3×K3K3 \times K3.

See also at C-field tadpole cancellation the section Integrality on K3×K3K3 \times K3.

Proposition

(Euler characteristic of K3×\timesK3)

The Euler characteristic of K3×K3K3 \times K3 is 24 224^2:

χ 8[K3×K3]=24 2. \chi_8[K3\times K3] \;=\; 24^2 \,.
Proof

By the Whitney sum formula for the Euler class we have χ 8[K3×K3]=(χ 4[K3]) 2\chi_8[K3 \times K3] = (\chi_4[K3])^2. Hence the statement follows by Prop. .

Proposition

(first Pontryagin class of K3×\timesK3)

The first Pontryagin class evaluated on K3×K3K3 \times K3 is:

p 1[K3×K3]=2×48 p_1[K3 \times K3] \;=\; - 2 \times 48
Proof

By the general formula for Pontryagin classes of product spaces we have

p 1(K3×K3) =p 0(K3)p 1(K3)+p 1(K3)p 0(K3) =2p 1(K3) \begin{aligned} p_1(K3 \times K3) & =\; p_0(K3) \smile p_1(K3) + p_1(K3) \smile p_0(K3) \\ & = 2 p_1(K3) \end{aligned}

With this, the statement follows by Prop. .

Of K3×X 4K3 \times X^4

Now consider the Cartesian product space K3×X 4K3 \times X^4 of K3 with some 4-manifold.

Proposition

(Euler characteristic of K3×X 4\times X^4)

The Euler characteristic of K3×K3K3 \times K3 is 2424 times the Euler characteristic of X 4X^4:

χ 8[K3×X 4]=24χ 4[X 4]. \chi_8[K3\times X^4] \;=\; 24 \cdot \chi_4[X^4] \,.
Proof

By the Whitney sum formula for the Euler class we have χ 8[K3×X 4]=χ 4[K3]χ 4[X 4]\chi_8[K3 \times X^4] = \chi_4[K3] \cdot \chi_4[X^4]. Hence the statement follows by Prop. .

Proposition

(first Pontryagin class of K3×X 4\times X^4)

The first Pontryagin class evaluated on K3×X 4K3 \times X^4 is:

p 1[K3×X 4]=p 1[X 4]48 p_1[K3 \times X^4] \;=\; p_1[X^4] - 48
Proof

By the general formula for Pontryagin classes of product spaces we have

p 1[K3×X 4] =p 0[K3]p 1[X 4]+p 1[K3]p 0[X 4] =p 1[X 4]+p 1[K3] \begin{aligned} p_1[K3 \times X^4] & =\; p_0[K3] \smile p_1[X^4] + p_1[K3] \smile p_0[X^4] \\ & = p_1[X^4] + p_1[K3] \end{aligned}

With this, the statement follows by Prop. .

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

As a fiber space in string compactifications

See

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 orbifold string theory:

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 November 11, 2019 at 07:08:46. See the history of this page for a list of all contributions to it.