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 surface over the complex numbers.
The algebro-geometric definition
- If is a smooth curve of degree and is the double cover of branched along , then surface; indeed in the Picard group of we have . A similar claim hods even if the curve is singular; the modification is that now one has to consider the normalization of the branch double cover. Specifically if the curve is a six lines tangent to a conic, then on recovers for the double cover model of a Kummer surface.
- A quartic surface in
- A complete intersection of a quadric and a cubic hyper-surfaces in
- A complete intersection of three quadric hypersurfaces in
In the last three examples one may verify that the canonical bundle is trivial using adjunction formula