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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

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The shaves of holomorphic sections of holomorphic vector bundles can be identified with the locally free sheaves. Any coherent sheaf admits a locally free resolution, and in this sense they are determined by vector bundles. –  Liviu Nicolaescu Mar 2 '13 at 17:50
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Dear John, coherent sheaves are locally cokernels of holomorphic maps between two holomorphic vector bundles. That is, $F$ is coherent if and only if locally on $X$, $F$ is the cokernel of a map $O_X^a \to O_X^b$ between two (trivial) vector bundles of finite ranks $a$, $b$. Hope this helps. –  Piotr Achinger Mar 2 '13 at 19:06
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Think of a complex submanifold $N\subset M$ and the trivial bundle $O_N$ on $N$, thought of as a sheaf on $M$. This is a coherent sheaf on $M$, but of course does not come from a vector bundle. Another example: the ideal sheaf of $N$ (sections of the trivial bundle $O_M$ on $M$ which vanish along $N$) will not be a vector bundle if $N$ has codimension $>1$, but will still be coherent. –  Piotr Achinger Mar 2 '13 at 19:08
    
@Liviu: Thanks for the comment. So if you take the derived category of the locally free sheaves, what would the corresponding subcategory of the Fukaya category under mirror symmetry? –  John McCarthy Mar 2 '13 at 20:00
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Locally free sheaves don't form an abelian category, and the abelian category "generated" by them is the category of coherent sheaves. –  Piotr Achinger Mar 2 '13 at 20:12
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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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