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.
Best Answer
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, $\operatorname{Hom}(\mathcal O_{\mathbf P^1}, \mathcal O_{\mathbf P^1}(-1) \oplus \mathcal O_{\mathbf P^1}(-1)) = 0$.