Let $X$ be a compact complex manifold and $\mathcal{O}_X$ be the structure sheaf of holomorphic functions. We call a sheaf of $\mathcal{O}_X$-module $\mathcal{F}$ coherent if it satisfies the following two conditions:
- $\mathcal{F}$ is locally finite generated: For each point $x\in X$ there exists an open neighborhood $U$ of $x$ and a surjective $\mathcal{O}_U$-module morphism $\mathcal{O}_U^{\oplus n}\twoheadrightarrow \mathcal{F}|_U$;
- For any open subset $V\subset X$ and any $\mathcal{O}_V$-module morphism $f: \mathcal{O}_V^{\oplus m}\twoheadrightarrow \mathcal{F}|_V$, its kernel $\ker(f)$ is locally finite generated.
Let $D^b_{coh}(X)$ denote the derived category of complexes of sheaves of $\mathcal{O}_X$-modules with bounded and coherent cohomologies. On the other hand let $D^b(coh(X))$ denote the derived category of bounded complexes of coherent $\mathcal{O}_X$-modules.
In Generators and representability of functors in commutative and noncommutative geometry Section 5.2 it was written that
If $X$ is algebraic then it is well-known and easy to prove that $D^b(coh(X))$ and $D^b_{coh}(X)$ are equivalent. We don’t know if the corresponding result is true for the complex analytic case.
My question is: if $X$ is a compact complex manifold, do we have $D^b_{coh}(X)\simeq D^b(coh(X))$?
