K3 surface

A K3 surface is a compact, complex two-dimensional manifold that is simply connected and admits a nowhere-vanishing holomorphic 2-form — making it the simplest non-trivial example of a Calabi-Yau manifold. Named in honor of Kummer, Kähler, and Kodaira (and the mountain K2), K3 surfaces occupy a privileged position in algebraic geometry: they are the only simply connected Calabi-Yau surfaces, and their Ricci-flat Kähler metrics make them critical testing grounds for ideas in string theory, where they appear as compactification geometries preserving some supersymmetry. The moduli space of K3 surfaces is 20-dimensional and admits a natural action of the orthogonal group O(3,19), revealing a deep connection between algebraic geometry, lattice theory, and arithmetic that continues to produce surprises at the boundary of what we know.