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There is a Serre dualizing complex $S_X\in D^b Coh(X)$ for any scheme $X$ of finite type. For proper schemes, it is characterized as the sheaf defining the right adjoint to derived global sections, and for a general scheme $U$, one can define $S_U: = j^*S_X$ for $j:U\to X$ the open embedding into some choice of compactification $X\supset U.$

Now given a scheme $X$ over $\mathbb{C},$ Serre's GAGA associates to it a ringed space $(X(\mathbb{C}), \mathbb{O}^{hol})$ of holomorphic sections on the topological space $X(\mathbb{C})$ with analytic topology. When $U$ is a smooth $n$-dimensional scheme, we have a canonical quasiisomorphism $$S_X \cong \omega_X[-n]$$ for the Serre dualizing complex in terms of the sheaf of top forms $\omega : = \Omega^n(X)$, which is an analytic invariant. I am interested in whether this continues to be true for singular schemes. I.e., I would like to know the following.

Question. Suppose $X$ is a scheme of finite type over $\mathbb{C}$ and $\underline{U}\subset X(\mathbb{C})$ is an analytic open. Is there an invariant of the ringed space $(\underline{U}, \mathbb{O}^{hol}\mid U)$ which recovers the holomorphic sections of the Serre dualizing sheaf $S_X^{hol}\mid \underline{U}$?

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    $\begingroup$ In Hartshorne's "Residues and Duality", there is a characterization of the (quasi-isomorphism class) of the stalks of the dualizing sheaf in terms of local cohomology. That should still work in the analytic category. Obviously when you choose $U$ to be all of $X$, the analytic and algebraic (bounded) derived categories of coherent are equal, so in that case you also have a characterization that works in the analytic category. This same question must arise for rigid analytic spaces, so perhaps you could search for dualizing complexes of rigid analytic spaces. $\endgroup$
    – Jason Starr
    Commented Jan 13, 2022 at 11:54
  • $\begingroup$ @JasonStarr Thank you! I had trouble finding the local cohomology identification (the pdf I have isn't easily searchable). But I'm pretty sure you're right. One argument to see this is to interpret the Serre dualizing complex using Verdier duality of holonomic D modules, which can be done in the analytic topology. $\endgroup$
    – Dmitry Vaintrob
    Commented Jan 14, 2022 at 12:57

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