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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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    $\begingroup$ 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. $\endgroup$ – Liviu Nicolaescu Mar 2 '13 at 17:50
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    $\begingroup$ 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. $\endgroup$ – Piotr Achinger Mar 2 '13 at 19:06
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    $\begingroup$ 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. $\endgroup$ – Piotr Achinger Mar 2 '13 at 19:08
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    $\begingroup$ Locally free sheaves don't form an abelian category, and the abelian category "generated" by them is the category of coherent sheaves. $\endgroup$ – Piotr Achinger Mar 2 '13 at 20:12
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    $\begingroup$ the use of coherent sheaves in several complex variables actually preceded its use in algebraic geometry. it was apparently introduced in the Cartan seminars in 1951-2, to systematize the work of K. Oka from the preceding 15 years. A textbook source is the classic by Gunning and Rossi from 1965, chapter 4. $\endgroup$ – roy smith Mar 5 '13 at 14:45
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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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