Example 4

\* Let X be a differential variety of class $C^r$ (with $r \geqslant 1$) pure of finite dimension $n$.

In the vector fibre space $\Lambda^n T(X)$, let L be the complement of the zero section. For every real number $\lambda \neq 0$, the mapping $u \mapsto \lambda u$ from $\Lambda^n T(X)$ into itself leaves L stable.

Let $\alpha$ be a complex number. A complex valued function $\omega$ on L such that $\omega(\lambda u) = |\lambda|^{\alpha} \omega(u)$ for $u \in L$ and any non-zero real number $\lambda$ is called a density of order $\alpha$ on X. We say that a density $\omega$ of order 1 is locally integrable if there exists an open cover $(U_i)_{i \in I}$ of X, and for every $i \in I$ a system of coordinates $\xi_i = (\xi_i^1, ..., \xi_i^n)$ on $U_i$ and a complex valued function $f_i$ on $\xi_i(U_i)$ satisfying the following conditions:

a) The function $f_i$ is locally integrable on the open set $\xi_i(U_i)$ of $\mathbf{R}^n$ with respect to the Lebesgue measure $\mu$;

b) Let $x \in U_i$; if $(\partial_{1,i,x}, ..., \partial_{n,i,x})$ is the basis of $T_x X$ associated to the system of coordinates $(\xi_i', ..., \xi_i^n)$ in $U_i$ we have

Then, there exists one and only one measure $\tilde{\omega}$ on X such that for every $i \in I$, the image under $\xi_i$ of the restriction of $\tilde{\omega}$ to $U_i$ is equal to the measure $f_i \cdot \mu$ (cf. VAR, R, 10.4.3).

Let $\mathscr{V}$ (resp. $\mathscr{N}$) be the vector space of measurable densities $\omega$ of order 1/2 such that the measure associated with the density $|\omega|^2$ of order 1 is bounded (resp. null). Let $\omega_1$ and $\omega_2$ be in $\mathscr{V}$; then $\omega = \overline{\omega_1} \omega_2$ is a density of order 1, and the measure $\tilde{\omega}$ associated with $\omega$ is bounded; the number $\int_X \tilde{\omega}$ depends only on the classes $\dot{\omega}_1$ and $\dot{\omega}_2$ of $\omega_1$ and $\omega_2$ modulo $\mathscr{N}$ and is denoted by $\langle \omega_1 | \omega_2 \rangle$ or $\langle \dot{\omega}_1 | \dot{\omega}_2 \rangle$. Then the mapping $(\dot{\omega}_1, \dot{\omega}_2) \mapsto \langle \dot{\omega}_1 | \dot{\omega}_2 \rangle$ assigns a complex hilbertian space structure to the vector space $\Omega_{1/2}(X) = \mathscr{V}/\mathscr{N}$.