Suppose $X$ is a smooth quasi-projective variety over $\mathbb{C}$ and $Z$ a proper subscheme, there is a formal duality isomorphism (here we consider the Zariski topology) due to Hartshorne:


$$ tr: R^n\Gamma\_{Z}(\Omega\_X^n) \cong \mathbb{C} $$


given in proposition 5.2 of his paper,"On the de Rham cohomology of algebraic varieties." In our setting there is a really simple trace in the analytic topology given by integration.


$$ R^n\Gamma\_{cs}(\Omega\_X^n) \cong \mathbb{C} $$


where we are considering cohomology with compact support. 


Question: Do these traces satisfy compatibility under GAGA and the universal $\delta$ map


$$ R^n \Gamma \_{Z\_{an}}(\Omega\_{X\_{an}}^n) \to R^n \Gamma\_{cs}(\Omega\_X^n) $$ I'd draw the diagram I want but I'm too incompetent. On the other hand I expect it's clear from the question. 



Comparison of two traces
The issue of equality of maps in the derived category is always a delicate issue. In the algebraic setting the commutativity is explained in Lipman's Asterisque 117. When you get into the analytic setting, then there is a comparison between the algebraic map and the one coming from De Rham theory that involves a *period*, namely $\frac{1}{2 \pi i}$.


As a matter of fact, when comparing the algebraic and the analytic traces, a sign arises that depends on the dimension of $X$. This is explained carefully in:


Sastry, Pramathanath; Tong, Yue Lin L.:
The Grothendieck trace and the de Rham integral. 
*Canad. Math. Bull.* **46** (2003), no. 3, 429–440. 


