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Given a smooth quasi-projective variety $X$ over $\mathbb{C}$ and bounded complexes of vector bundles $(P,d)$ and $(P',d')$ with compactly supported cohomology. It is well-known that such complexes satisfy Serre-duality. The standard proof that I have heard is to complete $X$ to a projective variety $\bar{X}$.

For me the intuition behind Serre duality is integration, and the intuition for the above result is that one can always integrate compactly supported differential forms. Unfortunately, I've never seen a place where this result is actually proven in this way.

Question: Is there a reference which proves Serre duality using compactly supported Dolbeault cohomology?

The proof I have in mind is the standard proof of Serre duality for projective varieties, but I remember in Serre's original paper there are some tricky points of topology on Frechet spaces which are complicated. I haven't actually seen the compactly supported Dolbeault theory used in any other papers since then, so I'm wondering whether there is any newer reference with the above statement and which covers its properties more systematically.

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Looks like this question fell in between the cracks. But just in case anyone is still interested in an answer, a fairly detailed proof of Serre's duality theorem, which applies to cohomologies with compact supports (including the Dolbeault complex, and other complexes of differential operators) can be found in Sec.5.1 of

Tarkhanov, N. N., Complexes of differential operators. Revised and updated translation from the Russian by P. M. Gauthier, Mathematics and its Applications (Dordrecht). 340. Dordrecht: Kluwer Academic Publishers. xviii, 396 p. (1995). ZBL0852.58076.

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