Riemann bilinear criterion for a period matrix ($\Omega Q^{-1}{\Omega }^t=0,-i\Omega Q^{-1}{\bar{\Omega }}^t>0$)

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 $\omega (v,Jv)>0$.