such that $$\tfrac{1}{i} {\omega}( x, \overline
{x}) >0$$ for any non-vanishing $x \in E$. Any positive polarization has a canonical Hermitian scalar product given by $$( x, y) \rightarrow \tfrac{1}{i} {\omega}(x, \overline{
  y}) .$$ The set of positive polarizations of $S$ is a contractible topological space.

For any positive polarizations, we consider the canonical line $\wedge^{{\operatorname{top}}} E^* $ with its associated Hermitian product. For any positive polarizations $E_a$ and $E_b$, $S \otimes {{\mathbb{C}}}$ is the direct sum of $ E_a$ and $\overline{E}_{b}$. Let $\pi_{E_b, E_a}: E_{b} \rightarrow E_{a}$ be the restriction of the projection onto $ E_{a}$ with kernel $\overline{E}_{b}$. This map is invertible. Define $$\Psi_{E_a, E_b} = \pi_{E_b, E_a} ^* : \wedge^{{\operatorname{top}}} E^*_{a}
\rightarrow  \wedge ^{{\operatorname{top}}} E^*_{b}$$ Given three positive polarizations, let $\zeta( E_a, E_b,
E_c)$ be the complex number such that $$\begin{gathered}
  \label{eq:def_zeta}
 \Psi _{ E_a, E_c} = \zeta( E_a, E_b, E_c) \Psi_{ E_b, E_c} \circ
\Psi_{ E_a, E_b}\end{gathered}$$ Define $\zeta^{1/2}( E_a, E_b , E_c)$ to be the square root depending continuously on $E_a$, $E_b$ and $E_c$ and taking the value 1 when $E_a = E_b = E_c$.

Half-form lines
---------------

For any positive polarization $E$, a half-form line of $E$ is a complex line $\delta$ together with 