I start my question with a definition and some motivation.

Let $M$ be a symplectic manifold. A subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if

$P$ is Lagrangian, i.e. Maximal isotroic, dim$P_m=n$, $\forall m\in M$, and $\omega(P,P)=0$

P involutive, i.e. if $X,Y\in P$ then $[X,Y]\in P$

dim$P\cap\bar P \cap TM$ is constant

Now, Introduce an hermitian form on $P$ defined by

$$b(X,Y)=i\omega(X,\bar Y).$$

Note that if $P$ is real, then $b$ is vanishes identically on $P$. Consequently, $b$ projects onto a non-degenerate form on the quotient $P/{(P\cap \bar P)}$ and we denote it by $\bar b$. $P$ is said to be of type $(r,s)$ if and only if $\bar b$ has signature $(r,s)$ i.e. its matrix is $$diag(\underbrace{1,1...,1}_{r},\underbrace{-1,-1...,-1}_{s} )$$ for $0\le r+s=n-dim_{\mathbf C }P\cap \bar P$ . Then, $P$ is said to be positive if $s=0$. In the case if $r=s=0$ then $P$ is real

Let $(M,\omega, J)$ be a compact Kahler manifold with positive-definite polarization $P$, and $(L,\nabla)$ be prequantum data. Let $$M_{quantum} = \left\{s\in \Gamma(L) \vert \nabla_Xs=0 , \forall X\in \bar P \right\}$$ Then I can not see why $M_{quantum}$ is fnite-dimensional?. I am looking for a referrence for finding a proof for this assertion "the space of square integrable holomorphic sections is closed."

Lagrangianandinvolutive? – Liviu Nicolaescu Dec 31 '13 at 14:31