Contents

complex geometry

# Contents

## Definition

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

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

## Properties

### 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 $1, 0, 22, 0, 1$:

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

### 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-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

### 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: