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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.

The term “K3” is (Weil 79, p. 546)

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

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, such as the Fermat quartic? {[w,x,y,z] 3|w 4+x 4+y 4+z 4=0}\{[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).

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

  • 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 Scientic works. Collected papers. Vol. II (1951-1964). 1979.

Lecture notes 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 20, 2015 06:40:19 by Urs Schreiber (94.118.135.251)