Let $\mathcal{F}$ be a coherent sheaf on a projective manifold $X$. It is well known that one can construct a resolution of $\mathcal{F}$ by holomorphic vector bundles (locally free sheaves).


Are two such resolutions homotopic? Any reference would be much appreciated.



When are two resolutions of a coherent sheaf homotopic
If this were true, then any short exact sequence of vector bundles would split. Indeed, if $0 \to \mathscr E\_1 \to \mathscr E\_2 \to \mathscr E\_3 \to 0$ is a short exact sequence of vector bundles, then both
\begin{align\*}
K^\bullet = \cdots \to 0 \to \mathscr E\_1 \to \mathscr E\_2 \to 0 \to \cdots
\end{align\*}
and $L^\bullet = \mathscr E\_3[0]$ are resolutions of $\mathscr E\_3$. If $g \colon L^\bullet \to K^\bullet$ is a homotopy equivalence (or even a quasi-isomorphism!), then the map $g^0 \colon \mathscr E\_3 \to \mathscr E\_2$ induces an isomorphism
$$\phi \colon \mathscr E\_3 \stackrel\sim\to H^0(K^\bullet) = \mathscr E\_3.$$
Then $g^0 \circ \phi^{-1} \colon \mathscr E\_3 \to \mathscr E\_2$ is a splitting of $0 \to \mathscr E\_1 \to \mathscr E\_2 \to \mathscr E\_3 \to 0$. $\square$


(On the other hand, there does always exist a quasi-isomorphism $K^\bullet \to L^\bullet$ in this case, just not in the other direction.)


An example of a short exact sequence of vector bundles that doesn't split is the Koszul sequence
$$0 \to \mathcal O\_{\mathbf P^1}(-2) \stackrel{\left(\begin{smallmatrix}-y \\ x\end{smallmatrix}\right)}\longrightarrow \mathcal O\_{\mathbf P^1}(-1) \oplus \mathcal O\_{\mathbf P^1}(-1) \stackrel{\left(x\ \ y\right)}\longrightarrow \mathcal O\_{\mathbf P^1} \to 0$$
on $X = \mathbf P^1$. Indeed, $\operatorname{Hom}(\mathcal O\_{\mathbf P^1}, \mathcal O\_{\mathbf P^1}(-1) \oplus \mathcal O\_{\mathbf P^1}(-1)) = 0$.


