geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
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
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)
All K3 surfaces are simply connected.
Over the complex numbers they are all Kähler, and even hyperkähler.
(integral cohomology of K3-surface)
The integral cohomology of a K3-surface $X$ is
(e.g. Barth-Peters-Van den Ven 84, VIII Prop. 3.2)
(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 $1, 0, 22, 0, 1$:
(e.g. Barth-Peters-Van den Ven 84, VIII Prop. 3.3)
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 $X$
See duality between heterotic and type II string theory
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
Lecture notes include
Daniel Huybrechts, Lectures on K3-surfaces (pdf)
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
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
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:
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:
Last revised on May 18, 2019 at 13:26:44. See the history of this page for a list of all contributions to it.