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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8$\begingroup$ 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 NicolaescuMar 2, 2013 at 17:50
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5$\begingroup$ 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 AchingerMar 2, 2013 at 19:06
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1$\begingroup$ 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 AchingerMar 2, 2013 at 19:08
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9$\begingroup$ Locally free sheaves don't form an abelian category, and the abelian category "generated" by them is the category of coherent sheaves. $\endgroup$– Piotr AchingerMar 2, 2013 at 20:12
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1$\begingroup$ 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 smithMar 5, 2013 at 14:45
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2 Answers
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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 https://arxiv.org/abs/math/0509284
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I think the general idea of "coherent sheaves are what you get what you try to make the category of vector bundles into an abelian category" is a good intuition to have, and this is true for arbitrary ringed spaces. Since, however, your question specifically mentions complex manifolds, let me say a few words about what happens in this setting, since there is some subtlety in what "the" category of coherent sheaves might even mean.
The relationship between coherent sheaves and vector bundles is much more delicate in the holomorphic case than in the world of algebraic geometry. For example, if $X$ is a (smooth Noetherian) scheme, then any coherent sheaf on $X$ can be resolved by (i.e. is quasi-isomorphic to) a complex of vector bundles. The analogous fact in the holomorphic world is not true: there exist "many" coherent analytic sheaves for which we cannot resolve by vector bundles (a famous example of such a coherent sheaf is due to Voisin). This leads to a lot of problems, and is indirectly related to other similar non-existence theorems (e.g. holomorphic vector bundles "rarely" admit holomorphic connections).
One specific example of this is when you try to define what "the" category of coherent analytic sheaves should be. As is standard in geometry, we want to say that this is given by the derived category of the category of bounded cochain complexes of coherent sheaves, denoted $D^\mathrm{b}(\mathsf{Coh}(X))$. However, the category that seems to arise more often "in practice" is that of coherent cohomology, $D_\mathrm{coh}^\mathrm{b}(\mathsf{Sh}(X))$, which is the subcategory of $D^\mathrm{b}(\mathsf{Sh}(X))$ spanned by complexes whose internal cohomology (i.e. kernel of the differential quotiented by the image of the previous differential) is a coherent sheaf in each degree. Again, in the algebraic case, if $X$ is a Noetherian scheme, then these two categories are equivalent: $D^\mathrm{b}(\mathsf{Coh}(X))\simeq D_\mathrm{coh}^\mathrm{b}(\mathsf{Sh}(X))$ (this is found in SGA 6, §II, Corollaire 2.2.2.1, for example); but in the holomorphic case, whether or not these two categories are equivalent is, as far as I am aware, an open question (apart from the case where $X$ is a compact surface, in which case they are known to indeed be equivalent).
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$\begingroup$ Those two categories are not equivalent in case of a very general 3-dimensional complex torus (and are probably not equivalent for any complex variety with "wrong" Picard group, i. e. not satisfying Lefschetz). This result is not yet published. It's somewhat independent result of A. Bondal and myself; I'm currently working on description of Verdier quotient of bigger category by the image of inclusion. $\endgroup$– Denis TDec 1, 2022 at 10:30
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$\begingroup$ Oh, this is exactly the setting in which Voisin builds the example of a coherent sheaf with no locally free resolutions — I wonder if this is related. But either way, I am very excited to read about this. Is there a preprint available? $\endgroup$– TimDec 1, 2022 at 10:34
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$\begingroup$ No, this is is strongly unpublished, i. e. still in filling the gaps and writing down state. Of course, this is based on Voisin's example of non-resolvable sheaf. $\endgroup$– Denis TDec 1, 2022 at 11:27