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It is well known that any coherent sheaf on a complex manifold (or more generally on some complex spaces) admits locally a resolution with locally free sheaves. It is also well known that for non-algebraic manifolds in general there is no global resolution.

Now my question: is it some kind of invariant that prevents the local resolutions to glue to a global one?

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    $\begingroup$ this is a great question! a "high-level" answer would be that you have a global resolution when the associated twisting cochain (à la Toledo–Tong) is concentrated in degrees $0$ and $1$, which means that you could maybe consider the higher degrees as obstruction data $\endgroup$
    – Tim
    Commented Aug 20, 2021 at 21:54
  • $\begingroup$ Dear @Andrei Halanay, would you mind giving a reference for the existence of local resolution with locally free sheaves on complex spaces? I have only seen it for complex manifolds. $\endgroup$
    – Doug Liu
    Commented Nov 22, 2023 at 17:51
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    $\begingroup$ Dear @Andrei Halanay, I agree that for every coherent module $F$ on a ringed space $X$ and every $x\in X$, there is an open neighbourhood $U$ of $x\in X$ such that $F|_U$ is can be generated by finitely many sections, i.e., there is an integer $n\ge0$ and a quotient $\phi:O_U^n\to F|_U$. Moreover, the kernel of $\phi$ is of finite type. However, $\ker(\phi)$ may not be generated by finitely many sections over $U$. To get a quotient like $O^m\to \ker(\phi)$, we have to shrink $U$. As we may not have a finite-length resolution with finite locally free sheaves, we need infinite shrinkings. $\endgroup$
    – Doug Liu
    Commented Nov 23, 2023 at 7:12
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    $\begingroup$ @Tim reading the question, I knew you would be interested! $\endgroup$
    – DamienC
    Commented Dec 7 at 21:19
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    $\begingroup$ You are probably aware of that, but Schuster in Locally free resolutions of coherent sheaves on surfaces has proven that for smooth compact surfaces global vector bundle resolutions exist. The proof uses classification. As far as I am aware, no such invariant is known, though people have been looking for it. $\endgroup$
    – Grisha Papayanov
    Commented Dec 8 at 15:33

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For the sake of trying to offer some sort of answer based off of Tim's first comment:

Let $\mathcal{E}$ be a coherent sheaf on $M$ and $(((E_i, \nabla_i)\to U_i \subset M_i)_{i \in I}$ be a local resolution of $\mathcal{E}$ subordinate to cover $\mathcal{U}= \{U_i\}_I$ with chosen connections and $g_{\bullet}$ a twisting cochain a la O'Brian, Toledo. and Tong. Define for each $(\alpha_0, \ldots, \alpha_p)$ the element $c^p_{(\alpha_0, \ldots, \alpha_p)}:= (\nabla g)^p_{(\alpha_0, \ldots, \alpha_p)} \in \check{C}^p(\mathcal{U}, \Omega_{hol}^p(\mathcal{U}, Hom^{\bullet}(E_{\alpha_p}, E_{\alpha_0})))$.

The collection of such elements for each $p$, $c^p_{\bullet}$ is Čech-Hom closed (I think this goes back to Atiyah, but we spelled out the details in GMTZ Prop 3.11) and we can put the zero differential on $\Omega^{\bullet}$ so that these elements arrange to form classes in a bicomplex. Alternatively, you can strip the hom-differential portion of the Čech complex out and say that these elements have components in a different bicomplex $c^p = \sum_{q = 0}^{\infty}c^{p,q}\in \check{C}^p(\mathcal{U}, \Omega_{hol}^p(\mathcal{U}, Hom^{q}(E_{\alpha_p}, E_{\alpha_0})))$.

Now since equivalent coherent sheaves induce cohomologous characteristic classes ``$tr_g(c^{\bullet})$'', we hopefully can rework those statements to obtain invariants in (at worst) this Čech-Hom space or maybe (at best) Čech-Forms space so that

  1. Equivalent coherent sheaves induces cohomologous classes, and
  2. A vector bundle thought of as a coherent sheaf has $c^{p,q}=0$ for $q< 0$.
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    $\begingroup$ Thanks! I think this is a really promising sketch of an idea — hopefully somebody works out the details in 2025 ;-) $\endgroup$
    – Tim
    Commented 12 hours ago

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