K3 surface

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

The complex geometry definition is a complete smooth simply connected surface with trivial canonical class. The algebro-geometric definition is projective smooth varieties with trivial first cohomology of the structure sheaf.