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). 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 (Weil 79, p. 546)
in honor of Kummer, Kähler, Kodaira, and the beautiful K2 mountain in Kashmir
A cyclic cover $\mathbb{P}^2$ 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).
All K3 surfaces are simply connected.
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 $1, 0, 22, 0, 1$.
Over the complex numbers they are all Kähler, and even hyperkähler?.
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 $X$
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
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