nLab K3 surface

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

(Weil 79, p. 546)

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 K3-surface

2. 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.

Homotopy

• All K3 surfaces are simply connected.

Cohomology

Proposition

(integral cohomology of K3-surface)

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

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

(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 $E_1$. This also implies that the Betti numbers are completely determined as $1, 0, 22, 0, 1$:

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

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

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:

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

(LLP 17, Lemma 1.5, Example 3.1, CLP 19, Theorem 13.5a, see at Calabi-Yau manifolds in SU-bordism theory)

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:

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

(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:

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

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

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

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

$n$	Calabi-Yau n-fold	line n-bundle	moduli of line n-bundles	moduli of flat/degree-0 n-bundles	Artin-Mazur formal group of deformation moduli of line n-bundles	complex oriented cohomology theory	modular functor/self-dual higher gauge theory of higher dimensional Chern-Simons theory
$n = 0$		unit in structure sheaf	multiplicative group/group of units		formal multiplicative group	complex K-theory
$n = 1$	elliptic curve	line bundle	Picard group/Picard scheme	Jacobian	formal Picard group	elliptic cohomology	3d Chern-Simons theory/WZW model
$n = 2$	K3 surface	line 2-bundle	Brauer group	intermediate Jacobian	formal Brauer group	K3 cohomology
$n = 3$	Calabi-Yau 3-fold	line 3-bundle		intermediate Jacobian		CY3 cohomology	7d 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).

Elliptic fibration

See at elliptically fibered K3-surface.

As a fiber space in string compactifications

See

• duality between heterotic and type II string theory

• F-theory on K3

• M-theory on Sp(1)-manifolds

Related concepts

• formal Brauer group

• elliptic fibration, elliptically fibered K3-surface

• K3-cohomology

• F-theory, F-theory on K3

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:

• Daniel Huybrechts, Lectures on K3-surfaces, Cambridge University Press 2016 (pdf, pdf, doi:10.1017/CBO9781316594193)

• Claire Voisin, Geometry of the Moduli Space of K3 surfaces

• David Morrison, The geometry of K3 surfaces Lecture notes (1988)

• Viacheslav Nikulin, Elliptic fibrations on K3 surfaces (arXiv:1010.3904)

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

• Michael Artin, Barry Mazur, Formal Groups Arising from Algebraic Varieties, Annales scientifiques de l’École Normale Supérieure, Sér. 4, 10 no. 1 (1977), p. 87-131 numdam, MR56:15663

Systematic construction of Ricci flat Riemannian metrics on K3 orbifolds:

• Shamit Kachru, Arnav Tripathy, Max Zimet, K3 metrics (arXiv:2006.02435)

• Arnav Tripathy, Max Zimet, A plethora of K3 metrics (arXiv:2010.12581)

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

• Chris Hull, Paul Townsend, section 6 of Unity of Superstring Dualities, Nucl.Phys.B438:109-137,1995 (arXiv:hep-th/9410167)

• Edward Witten, section 4 of String Theory Dynamics In Various Dimensions, Nucl.Phys.B443:85-126,1995 (arXiv:hep-th/9503124)

Review includes

• Paul Aspinwall, K3 Surfaces and String Duality, in Shing-Tung Yau (ed.): Differential geometry inspired by string theory 1-95 (arXiv:9611137, spire:426102)

Further discussion includes

• Paul Aspinwall, David Morrison, String Theory on K3 Surfaces, in Brian Greene, Shing-Tung Yau (eds.), Mirror Symmetry II, International Press, Cambridge, 1997, pp. 703-716 (arXiv:hep-th/9404151)

• Paul Aspinwall, Enhanced Gauge Symmetries and K3 Surfaces, Phys.Lett. B357 (1995) 329-334 (arXiv:hep-th/9507012)

Specifically in relation to orbifold string theory:

• Katrin Wendland, Orbifold Constructions of K3: A Link between Conformal Field Theory and Geometry, in Orbifolds in Mathematics and Physics [arXiv:hep-th/0112006]

• Anne Taormina, Katrin Wendland, Symmetry-surfing the moduli space of Kummer K3s, Proc. Symp. Pure Math. 90 (2015) 129 [arXiv:1303.2931, doi:10.1090/pspum/090/01522]

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

• Inaki Garcia-Etxebarria, Angel Uranga, From F/M-theory to K-theory and back, JHEP 0602:008,2006 (arXiv:hep-th/0510073)

Specifically in M-theory on G₂-manifolds:

• Michael Atiyah, Edward Witten section 6.4 of $M$-Theory dynamics on a manifold of $G_2$-holonomy, Adv. Theor. Math. Phys. 6 (2001) (arXiv:hep-th/0107177)

Specifically in relation to Moonshine:

• Miranda Cheng, Sarah M. Harrison, Roberto Volpato, Max Zimet, K3 String Theory, Lattices and Moonshine (arXiv:1612.04404)

Specifically in relation to little string theory:

• Shamit Kachru, Arnav Tripathy, Max Zimet, K3 metrics from little string theory (arXiv:1810.10540)