geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
A K3 surface is a Calabi-Yau variety of dimension $2$. This means that the canonical bundle $\omega_X=\wedge^2\Omega_X\simeq \mathcal{O}_X$ is trivial and $H^1(X, \mathcal{O}_X)=0$.
A cyclic cover $\mathbb{P}^2$ branched over a curve of degree $6$
A nonsingular degree $4$ hypersurface in $\mathbb{P}^3$.
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.
David Morrison, The geometry of K3 surfaces Lecture notes (1988)
Viacheslav Nikulin, Elliptic fibrations on K3 surfaces (arXiv:1010.3904)