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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.

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  • $\begingroup$ In fact , in Kahler polarization case, we can construct a non-degenerate hermitian metric on $(M,\Omega , J)$ in the following $$K(X,Y)=g(X,Y)-i\Omega(X,Y)$$ where $g$ is a pseudo Riemannian metric .Then we can say that $k$is positive iff $\mathbb{P}$ be positive, i.e $\mathbb{P}$ is a kahler polarization of type $(r,s)$ with $s=0$. But Kahler polarization is not my interest in this question $\endgroup$
    – user21574
    Nov 4, 2013 at 15:58

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