Thanks! This is great. Looking at Aaron Smith's website, I saw he's working on a paper with Block on a constructible Riemann-Hilbert correspondence.

Thank you! I looked at the paper. In terms of its notation, are you saying that the horizontal section is the 1-eigenspace of the derivative of the fundamental endomorphism?

Thanks! Yes, this is exactly what I need. Do you have a link for Foulon's thesis?

Thanks Sasha! Neat. I agree that when coherent cohomology is trivial, we get no information. However are you sure that there's no data if only $H^n(O)=0$? For example what if we compose your first map with the multiplication map $Hom(O_X,O_S)\otimes Hom(O_X, O_S)\to Hom(O_X, O_S)$ to get a map $Hom(O_X,O_S)\otimes Hom(O_X,O_S)\otimes Ext^n(O_S,O_X)\to Ext^n(O_X,O_X)$. Then if $H^n(O_X)=0$ then this product will of course be zero, but there will be a well-defined and possibly nonzero Massey product. Is it clear that this Massey product will be zero?

Hm... I agree that it seems like something is lost in dimension $>1$, but not sure your argument is correct, or else I'm not following it. On a curve, if you take the extension closed abelian subcategory of $(D^b)Coh(X)$ generated by $O_X$ and all structure sheaves of points, you get all of $Coh$, which certainly lets you recover $X$. So despite the fact that Homs and Exts between objects in the subcategory <i>consisting</i> of $O_X$ and skyscraper sheaves depend only on the coherent cohomology, once you start taking extensions things can become more subtle. How does Hartog change this?

Yes, of course (otherwise it'd be boring in dimension $>1$).