Mathematics – Algebraic Geometry
Scientific paper
2003-01-14
Mathematics
Algebraic Geometry
35 pages
Scientific paper
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.
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