Une caract\'erisation diff\'erentielle des faisceaux analytiques coh\'erents sur une vari\'et\'e complexe

We give a generalization, in the context of sheaves, of a classical result of Grothendieck concerning the integrability of connections of type $(0,1)$ over a ${\mathcal{C}}^{\mathrm{\infty }}$ vector bundle over a complex manifold. We introduce the notion of $\bar{\mathrm{\partial }}$-coherent sheaf, which is a ${\mathcal{C}}^{\mathrm{\infty }}$ notion, and we prove the existence of an (exact) equivalence between the category of coherent analytic sheaves and the category of $\bar{\mathrm{\partial }}$-coherent sheaves. The principal difficulty of the proof is the solution of a quasi-linear differential equation with standard $\bar{\mathrm{\partial }}$ as its principal term. We are able to find a solution of this differential equation, using a rapidly convergent iteration scheme of Nash-Moser type.