Une caractérisation différentielle des faisceaux analytiques cohérents sur une variété complexe

Abstract

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 ${\mathcal{C}}^{\infty }$ vector bundle over a complex manifold. We introduce the notion of $\bar{\partial }$-coherent sheaf, which is a ${\mathcal{C}}^{\infty }$ 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.

35 pages