= Riemann bilinear criterion for a period matrix
{c}
{title2=$\Omega Q^{-1}\Omega^t=0,\quad-i\Omega Q^{-1}\bar\Omega^t>0$}
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 $\omega(v,Jv)>0$.
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