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Apr 18, 2019 at 16:06 comment added user137767 @pbelmans I think it is somewhat unfair from the historical point of view (because from my understanding, the field of shifted symplectic geometry came into existence because of such examples). For this particular case, it is probably enough to understand Serre duality well. Maybe shifted symplectic geometry provides a uniform point of view on such examples but I do not think that (at this point of time) it provides such a deep perspective that we would call it an explanation of this example. Would be happy if somebody explains whether/why I am wrong.
Apr 18, 2019 at 13:19 comment added pbelmans Isn't this miracle in part explained by (or at least closely related to) results in shifted symplectic geometry, where the shiftedness is given by $2-n$ where $n$ is the dimension, so for surfaces one ends up with ordinary symplectic objects?
Apr 18, 2019 at 12:55 history answered Libli CC BY-SA 4.0