nLab
K3 surface

Contents

Definition

A K3 surface is a Calabi-Yau variety of dimension 22 (a Calabi-Yau algebraic 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.

Examples

  • A cyclic cover 2\mathbb{P}^2 branched over a curve of degree 66

  • A nonsingular degree 44 hypersurface in 3\mathbb{P}^3.

Properties

Basic properties

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-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=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

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)