Compact Riemann surface
= 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>.