K3 surface

In complex geometry and in algebraic geometry K3 surfaces are the 2-dimensional analog of elliptic curves. The complex and algebro-geometric definitions are slightly different, and coincide in the case where the surface is an algebraic $K3$ surface over the complex numbers.

The algebro-geometric definition

In algebraic geometry a surface $S$ is a $K3$ surface if it is smooth, projective, with trivial canonical bundle, and such that $h^{1}(O_{S})=0$. In this case one automatically gets: $h^{2}(O_{S})=1$.

Examples

• If $C\subset \mathbb {P} ^{2}$ is a smooth curve of degree $6$ and $p:S\to \mathbb {P} ^{1}$ is the double cover of $\mathbb {P} ^{2}$ branched along $C$, then $S$ $K3$ surface; indeed in the Picard group of $S$ we have $K_{S}=p^{*}(K_{\mathbb {P} ^{2}}-{\frac {1}{2}}[C])=p^{*}0=0$. A similar claim hods even if the curve $C$ is singular; the modification is that now one has to consider the normalization of the branch double cover. Specifically if the curve $C$ is a six lines tangent to a conic, then on recovers for the double cover model of a Kummer surface.
• A quartic surface in $\mathbb {P} ^{3}$
• A complete intersection of a quadric and a cubic hyper-surfaces in $\mathbb {P} ^{4}$
• A complete intersection of three quadric hypersurfaces in $\mathbb {P} ^{5}$

In the last three examples one may verify that the canonical bundle is trivial using adjunction formula

Complex definition

In complex geometry a surface is complete smooth simply connected surface with trivial canonical class.