Compact Riemann surface

A Riemann surface which is compact as a topological space. For a connected one of genus $g$, the Riemann-Roch theorem gives $g$ independent holomorphic differential forms. The Riemann bilinear relations for a compact surface make their normalized periods a full lattice in ${\mathbb{C}}^g$, and the Abel-Jacobi map of a compact Riemann surface turns this lattice into a complex torus encoding divisor classes.