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.

On $\mathbb{C}/(\mathbb{Z}+\mathbb{Z}i)$, a Hermitian form linear in its first argument has integral imaginary part on the period lattice exactly when its real scalar coefficient is integral. Positivity is equivalent to $t>0$. The associated holomorphic line bundle has degree $t$ because its First Chern class is represented by $(it/2)dz\wedge d\overline{z}$. In particular the oriented Chern pairing is minus the imaginary part under this convention.

A complex torus with period matrix of a complex torus $\Omega$ admits a Hodge metric precisely when there is a nonsingular integral skew-symmetric matrix $Q$ satisfying the displayed conditions. The entries of $Q$ are the periods of the invariant Kähler form over lattice two-tori. The first condition imposes type $(1,1)$, and the second imposes positivity; the factor $-i$ corresponds to the convention $\omega (v,Jv)>0$.

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.