  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
${\cal C}^{\infty}$ vector bundle over a complex manifold. We introduce the
notion of $\bar{\partial}$-coherent sheaf, which is a ${\cal C}^{\infty}$
notion, and we prove the existence of an (exact) equivalence between the
category of coherent analytic sheaves and the category of
$\bar{\partial}$-coherent sheaves. The principal difficulty of the proof is the
solution of a quasi-linear differential equation with standard $\bar{\partial}$
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.
