Riemann form on a complex torus ($H$)

A Riemann form on $V/\Gamma$ is a Hermitian form $H$ whose imaginary part is integer-valued on $\Gamma \times \Gamma$. Its imaginary part is equivalently an integral alternating form compatible with the complex structure.

A polarization of a complex torus is a positive Riemann form, equivalently the first Chern class of an ample line bundle. It is principal when its alternating form identifies the lattice with its dual.