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.
Categories: math.AG math.CV