K3 surface

Remarks

Over $\C$, the same two conditions define K3 surfaces among all compact complex surfaces; such a surface need not be projective. A K3 surface has Hodge numbers $h^{2,0} = 1$ and $h^{1,1} = 20$, second Betti number $22$ and topological Euler characteristic $24$. Example: a smooth quartic $X \subset \mathbb{P}^3$ has $\omega_X \cong \mathcal{O}_X(4 - 4) = \mathcal{O}_X$ by adjunction, and $H^1(X, \mathcal{O}_X) = 0$ from the exact sequence $0 \to \mathcal{O}_{\mathbb{P}^3}(-4) \to \mathcal{O}_{\mathbb{P}^3} \to \mathcal{O}_X \to 0$ and the vanishing of $H^1$ and $H^2$ of line bundles on $\mathbb{P}^3$. Non-example: an abelian surface, such as $E \times E$ for an elliptic curve $E$, has trivial canonical bundle but $\dim H^1(X, \mathcal{O}_X) = 2$, so it fails clause 2. Other examples are double covers of $\mathbb{P}^2$ branched along a smooth sextic and the Kummer surfaces, obtained from abelian surfaces by dividing by $x \mapsto -x$ and resolving the $16$ singular points.