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This is a more precise version of my previous question. Let $X$ be a smooth variety of dimension $n$ over $\mathbb{C}$ and $Z$ a proper sub-scheme. We denote by $\tilde{X}$ the formal completion of $X$ along $Z$. We have an isomorphism from page 22 of http://www.math.purdue.edu/~lipman/papers/formal-duality.pdf

$$ Ext^{n-i}_{\tilde{X}}(E,\omega_{\tilde{X}}) \cong (H^{\bullet}R\Gamma(\tilde{X},R\Gamma'_\tilde{X}(E))^i $$

See their paper for the definition of $R\Gamma'$.

Question: Is there some more involved version of residue integration which allows me to realize this map in an analytic fashion?

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