Construction of a resolution for a coherent sheaf

Question

Let $\mathcal{S}$ be a coherent sheaf over a complex manifold $M$. How do I construct a resolution of $\mathcal{S}$ by holomorphic vector bundles? Is this construction "unique"? Are the answers the same when $\mathcal{S}$ is a vector bundle over a subvariety?

I would be interested in references on the differential geometry side if there exists some (else any would be appreiated).

Thanks in advance

Answer 1

What you would want to do is to find a surjective morphism from a locally free sheaf $\mathcal{L}_0$ to $\mathcal{S}$, then find a surjective morphism from a locally free sheaf $\mathcal{L}_1$ to the kernel of $\mathcal{L}_0\to \mathcal{S}$, and repeat: this would give you a locally free resolution.

In general, you can't do this because of the following counterexample of Voisin:

On any generic complex torus of dimension at least 4, the ideal sheaf of a point does not admit a global locally free resolution.

(This is from her article "A Counterexample to the Hodge Conjecture Extended to Kahler Varieties", available here).

Even if you could do this in general, it is not true that such a resolution would be unique: just direct sum in a copy of the complex $0\to \mathcal{O}_X\stackrel{id}{\to}\mathcal{O}_X\to 0$ somehwere. What you would want to say is that these resolutions are unique up to chain homotopy or something like that, coming from the "fact" that a locally free sheaf is a projective object. But this "fact" is false, as seen in Piotr Achinger's example found here:

This is never true whenever $X$ has positive dimension. Let $L$ be ample on $X$ and let $E$ be a nonzero coherent sheaf on $X$. Let $P$ be any point of $X$ at which $E$ has a nonzero fiber, so we get a surjection $\mathcal{O}_X \to \mathcal{O}_P$ ($\mathcal{O}_P$ being the skyscraper sheaf at $P$). We can also find a $k>0$ such that the sheaf $\mathrm{Hom}(E, E\otimes L^{-k})= \mathrm{End}(E)\otimes L^{-k}$ has no nonzero global sections. Now tensor the surjection $\mathcal{O}_X\to \mathcal{O}_P$ by $E$ and $E\otimes L^{-k}$, getting surjections $a:E\to E_P$ and $b:E\otimes L^{-k}\to (E\otimes L^{-k})_P = E_P$. We cannot lift $a$ along $b$ because by assumption on $k$ there are no nonzero maps $E\to E\otimes L^{-k}$.

Nothing changes if you move to considering coherent sheaves which are supported on a proper submanifold.