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I am wondering if there is some relationship between Serre duality and Pontryagin duality for compact complex manifolds. In this case Serre duality reduces to the commutativity of Hodge-star operator with the Laplace operator. While differential operators can be written as pseudo-differential operators with symbols in the cotangent bundle, I do not know how this is related to the usual character theoretic characterization of Pontryagin duality.

A compact manifold is usually not a group, and there is no analgous Haar measure on it. Since Fourier inversion still holds for a suitable class of functions, I am wondering if there is any way to extend the Pontryagin duality to more "global" objects. Is it possible to incorporate the notion of cotangent bundle (or dualizing sheaf) in Pontryagin duality? Is it possible to interpret symbol of operators corresponding to elements in some "dual group"? I am not sure if this is a well-investigated topic, however after extensive literature search I did not find anything.

I am totally okay with a pure algebraic perspective without using analytical part of Hodge theory (say using Gronthendieck duality or something).

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    $\begingroup$ When I was in grad school, I took a topology class from John Morgan, where he explained how to formulate Poincaré duality with coefficients in a locally compact abelian group; the other side takes values in the Pontryagin dual. (I'm not sure it's written any place.) I can imagine a similar set up for Serre duality, but this seems different from what you're asking about. $\endgroup$
    – Donu Arapura
    Commented Aug 12, 2018 at 12:37
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    $\begingroup$ @DonuArapura in full generality it's probably not written down, but there's a version of this "Poincaré-Pontrjagin duality" in this paper by Freed-Teleman (e.g. just after (3.32)). $\endgroup$
    – Arun Debray
    Commented Aug 12, 2018 at 13:10
  • $\begingroup$ @DonuArapura: I am actually looking for something quite modest. As a first approximation, Poincare duality can be interpreted as cup product of differential forms is non-degenerate. Is there something similar using Fourier transform? I am not entirely sure what Fourier transform of a differential form should be, for example. $\endgroup$
    – Bombyx mori
    Commented Aug 12, 2018 at 23:37
  • $\begingroup$ @ArunDebray: I have read quite a bit on Ising model, and I have read some of Daniel Freed's papers. However the specific paper you quoted is unreadable to me. I am kind of lost with notation. So I am looking for a specific 2D example. $\endgroup$
    – Bombyx mori
    Commented Aug 12, 2018 at 23:39
  • $\begingroup$ What is "Gronthendieck duality"? $\endgroup$
    – Bernie
    Commented Aug 13, 2018 at 0:03

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