Definition 1

Let $\boldsymbol{a}= (a_1, . . . , a_n)\in A^n$, let $h:\mathbf{C}^n\rightarrow \mathbf{C}$ be a mapping and let $u_1, . . . , u_n$ be mappings of $\mathbf{C}^n$ into A. We say that the sequence $(h, u_1, . . . , u_n)$ is adapted to $\boldsymbol{a}$ if

(i) The mapping $h$ is infinitely differentiable, of compact support, and equal to 1 in the neighbourhood of Sp$^n(\boldsymbol{a})$ ;

(ii) The mappings $u_1, . . . ,u_n$ are infinitely differentiable ;

(iii) For every $\boldsymbol{z}= (z_1, . . . , z_n)\in \mathbf{C}^n$, one has

The differential form of degree $2n$ on $\mathbf{C}^n$, with coefficients in A, defined by

is called the differential form associated with $(h, u_1, . . . , u_n)$.

If $(h, u_1, . . . , u_n)$ is adapted to $\boldsymbol{a}$, then, by differentiating (1), one obtains the equality

whence, for every $i$ such that $1\leqslant i\leqslant n$, the relation