complex geometry

# Contents

## Definition

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

## Examples

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

• A nonsingular degree $4$ hypersurface in $\mathbb{P}^3$.

## Properties

### 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-Cau 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

## References

Original sources include

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

Revised on July 6, 2014 07:16:00 by Urs Schreiber (192.76.8.26)