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
I think not only can two such resolutions not be homotopy equivalent, but there might be no non-zero maps between them at all.
One way to construct a resolution of a coherent sheaf on a projective scheme is to use Serre's theorem. Take a sheaf $\mathcal{F}$ on a Noetherian projective scheme $X$, multiply it by $\mathcal{O}_X(n)$, for large enough $n$ it is globally generated, you surject on it from a direct sum of $\mathcal{O}_X$ and twist back. The beginning of your resolution then looks like $$\mathcal{O}_X^{\oplus s}(-n)\to\mathcal{F}\to0.$$
But if I take a greater twist, i.e. some $\mathcal{F}(m)$, $m>n$. I get a resolution starting with $$\mathcal{O}_X^{\oplus r}(-m)\to\mathcal{F}\to0.$$
But there are simply no non-zero morphisms from $\mathcal{O}_X^{\oplus s}(-n)$ to $\mathcal{O}_X^{\oplus r}(-m)$ in this case.
So no, I don't think there is a canonical choice of a locally free resolution in the homotopy category of coherent sheaves. There is, however, a canonical flat resolution on, I think, any scheme. A reference that was hugely useful for me is these notes by Daniel Murfet.