My question is about positivity of polarization in Geometric quantization theory. Let $\mathbb{P}$, be a complex polarization on symplectic manifold $(M,\Omega)$. For every $m\in M$, we can define a pseudo hermitian form $\widehat{h}_m$ in $\mathbb{P_m}$, by $$ \widehat{h}_m(X_m,Y_m)=i\Omega_m(X_m,\bar{Y_m}) $$ the fact is that $\widehat{h}_m$ projects onto a non-degenerate form on the quotient $\mathbb{P}/\mathbb{P}\cap \mathbb{\bar{P}}$. We denote this form by $\bar{h}$. Let the bilinear form $\bar{h}$ has the signature $(r,s)$, the we say the polarization $\mathbb{P}$ is positive, if $s=0$. So,
I am looking for a necessary and sufficient conditoin that the polarization $\mathbb{P}$ be positive.
PS:(Some motivation) In real Polarization, $\widehat{h}$ is always zero, so real polarization $\mathbb{P}$ is always positive.So that's why , I considered complex polarization.