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.


1 Answer 1


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.


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