A totally complex polarization of a symplectic manifold $(M,\omega)$ is a subbundle $F$ of the complexified tangent bundle $TM_{\mathbb C}:=TM\oplus iTM$ such that
- $F$ is integrable in the sense that Lie brackets of sections of $F$ are still sections of $F$,
- every fiber $F_x$ is a Langrangian subspace of $T_xM_{\mathbb C}$, i.e. $\dim_{\mathbb C}F_x=\dim M/2$ and $\omega(Z,W)=0$, $\forall Z,W\in F_x$,
- $F_x\cap\bar F_x=\{0\}$, $\forall x$ (and hence $F\oplus\bar F=TM_{\mathbb C}$).
It is said in some books or lectures on geometric quantization e.g. Bate and Weinstein's, there is a complex structure $J$ compatible with the symplectic form $\omega$ such that $F=\{X+iJX|\ X\in TM\}$. It is clear that $J$ has to be defined by multiplying by $-i$ on $F$ and by $i$ on $\bar F$. Then such $J$ preserves the real tangent bundle $TM$ and the symplectic form $\omega$. But I don't know how to prove that the symmmetric bilinear form $\omega(-,J-)$ is a positive definite metric on $TM$.
Moreover it seems to me that there is a simple 'counterexample' in $(\mathbb R^4,\omega:=dx^1\wedge dx^3+dx^2\wedge dx^4)$ where the totally complex polarization $F$ is spanned by vector fields $Z:=\partial/\partial x^1+i\partial/\partial x^2$ and $\partial/\partial x^4-i\partial/\partial x^3$. Then $J\partial/\partial x^1=J(\bar Z+Z)/2=i(\bar Z-Z)/2=\partial/\partial x^2$ hence $\omega(\partial/\partial x^1,J\partial/\partial x^1)=0$. So where is this 'counterexample' wrong?