For a complex manifold $M$, I'm trying to understand (from a differential geometry point of view) what its *category of coherent sheaves* is. If I understand correctly, then the sheaf of holomorphic sections of a complex vector bundle is a coherent sheaf for $M$. Is there some case when the category of coherent sheaves is equivalent to the category of holomorphic vector bundles? In other words can I just forget about the algebraic geometry and understand the definition in terms of complex geometry? Or alternatively, can one somehow *"generate"* the category of coherent sheaves from the category of holomorphic vector bundles

## 1 Answer

There are two approaches to this question, one by Toledo and Tong (using Cech covers or hypercovers) and one by Block (using the Dolbeaut algebra). Block's paper is at http://arxiv.org/abs/math/0509284

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