Contents

complex geometry

# Contents

## Definition

A K3 surface is a Calabi-Yau variety of dimension $2$ whose Picard variety is zero-dimensional. In other words, it is a complex algebraic surface with trivial canonical bundle ($\omega_X=\wedge^2\Omega_X\simeq \mathcal{O}_X$) and with $H^1(X, \mathcal{O}_X)=0$.

The term “K3” is

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

## Examples

• A cyclic cover of $\mathbb{P}^2$ branched over a curve of degree $6$.

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

• The flat orbifold quotient of the 4-torus (equipped with some complex structure) by the sign involution on all four canonical coordinates is the flat compact 4-dimensional orbifold known as a Kummer surface $T^4 \sslash \mathbb{Z}_2$, a singular K3-surface (e.g. Bettiol, Derdzinski & Piccione 2018, 5.5; Taormina & Wendland 2015, §1)

## Properties

### Compact hyperkähler structure

Over the complex numbers K3 surfaces are all Kähler, and even hyperkähler.

The only known examples of compact hyperkähler manifolds are Hilbert schemes of points $X^{[n+1]}$ (for $n \in \mathbb{N}$) for $X$ either

1. a 4-torus (in which case the compact hyperkähler manifolds is really the fiber of $(\mathbb{T}^4)^{[n]} \to \mathbb{T}^4$)

(Beauville 83) and two exceptional examples (O’Grady 99, O’Grady 03 ), see Sawon 04, Sec. 5.3.

### Cohomology

###### Proposition

(integral cohomology of K3-surface)

The integral cohomology of a K3-surface $X$ is

$H^n(X,\mathbb{Z}) \;\simeq\; \left\{ \array{ \mathbb{Z} &\vert& n = 0 \\ 0 &\vert& n = 1 \\ \mathbb{Z}^{22} &\vert& n = 2 \\ 0 &\vert& n = 3 \\ \mathbb{Z} &\vert& n = 4 } \right.$
###### 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 $E_1$. This also implies that the Betti numbers are completely determined as $1, 0, 22, 0, 1$:

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

### SU-Bordism

###### Proposition

(K3-surface spans SU-bordism ring in degree 4)

The canonical degree-4 generator $y_4 \in \Omega^{SU}_4$ in the SU-bordism ring (this Prop.) is represented by minus the class of any (non-torus) K3-surface:

$\Omega^{SU}_4 \;\simeq\; \mathbb{Z}\big[ \tfrac{1}{2}\big]\big\langle -[K3] \big\rangle \,.$

### Characteristic classes

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

#### Of $K3$

###### Proposition

(Euler characteristic of K3)

The Euler characteristic of K3 is 24:

$\chi_4[K3] \;=\; 24 \,.$
###### Proposition

(first Chern class of K3)

The first Chern class of K3 vanishes:

$c_1(K3) \;=\; 0 \,.$
###### Proposition

(second Chern class of K3)

The second Chern class of K3 evaluates to 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:

$p_1[K3] \;=\; -48 \,.$

###### 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:

\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 $K3 \times K3$

Now consider the Cartesian product space $K3 \times K3$.

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

###### Proposition

(Euler characteristic of K3$\times$K3)

The Euler characteristic of $K3 \times K3$ is $24^2$:

$\chi_8[K3\times K3] \;=\; 24^2 \,.$
###### Proof

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

###### Proposition

(first Pontryagin class of K3$\times$K3)

The first Pontryagin class evaluated on $K3 \times K3$ is:

$p_1[K3 \times K3] \;=\; - 2 \times 48$
###### Proof

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

\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 $K3 \times X^4$

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

###### Proposition

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

The Euler characteristic of $K3 \times K3$ is $24$ times the Euler characteristic of $X^4$:

$\chi_8[K3\times X^4] \;=\; 24 \cdot \chi_4[X^4] \,.$
###### Proof

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

###### Proposition

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

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

$p_1[K3 \times X^4] \;=\; p_1[X^4] - 48$
###### Proof

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

\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 $p$:

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 $h = \infty$, by definition;

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

moduli spaces of line n-bundles with connection on $n$-dimensional $X$

$n$Calabi-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
$n = 0$unit in structure sheafmultiplicative group/group of unitsformal multiplicative groupcomplex K-theory
$n = 1$elliptic curveline bundlePicard group/Picard schemeJacobianformal Picard groupelliptic cohomology3d Chern-Simons theory/WZW model
$n = 2$K3 surfaceline 2-bundleBrauer groupintermediate Jacobianformal Brauer groupK3 cohomology
$n = 3$Calabi-Yau 3-foldline 3-bundleintermediate JacobianCY3 cohomology7d Chern-Simons theory/M5-brane
$n$intermediate Jacobian

### Relation to third stably framed bordism group

The third stable homotopy group of spheres (the third stable stem) is the cyclic group of order 24:

$\array{ \pi_3^s &\simeq& \mathbb{Z}/24 \\ [h_{\mathbb{H}}] &\leftrightarrow& [1] }$

where the generator $[1] \in \mathbb{Z}/24$ is represented by the quaternionic Hopf fibration $S^7 \overset{h_{\mathbb{H}}}{\longrightarrow} S^4$.

Under the Pontrjagin-Thom isomorphism, identifying the stable homotopy groups of spheres with the bordism ring $\Omega^{fr}_\bullet$ of stably framed manifolds (see at MFr), this generator is represented by the 3-sphere (with its left-invariant framing induced from the identification with the Lie group SU(2) $\simeq$ Sp(1) )

$\array{ \pi_3^s & \simeq & \Omega_3^{fr} \\ [h_{\mathbb{H}}] & \leftrightarrow & [S^3] \,. }$

Moreover, the relation $2 4 [S^3] \,\simeq\, 0$ is represented by the bordism which is the complement of 24 open balls inside the K3-manifold (Wang-Xu 10, Sec. 2.6).

See

## References

### General

Original sources:

• 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:

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

Systematic construction of Ricci flat Riemannian metrics on K3 orbifolds:

### Elliptic fibrations

• O. Lecacheux, Weierstrass Equations for the Elliptic Fibrations of a K3 Surface In: Balakrishnan J., Folsom A., Lalín M., Manes M. (eds.) Research Directions in Number Theory Association for Women in Mathematics Series, vol 19. Springer (2019) (doi:10.1007/978-3-030-19478-9_4)

• Marie Bertin, Elliptic Fibrations on K3 surfaces, 2013 (pdf)

### 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 July 18, 2024 at 11:09:33. See the history of this page for a list of all contributions to it.