Analogy between BV formalism and integration by residues

Domenico Fiorenza begins his description of the Batalin–Vilkovisky formalism by pointing out an analogy with integration by residues:

• Take a top form (density) on $$\mathbf R$$ resp. space of fields;
• Double[*] the dimension of the space, adding imaginary numbers resp. antifields;
• Turn what used to be a density to be integrated over the whole space into a closed half-density to be integrated over a closed submanifold of half the total dimension;
• Use the closedness to choose a more convenient homologous cycle—for example, one that affords a power series expansion for the integral.

He doesn’t present this as anything more than skin deep, but I have to wonder—is there some actual mathematics here? Or maybe I’m just missing something obvious, like integration by residues relying on the symplectic structure on $$\mathbf C$$ (?) and the BV formalism being essentially odd symplectic geometry?

[*] Note, though, that integration by residues adds even coordinates while BV adds odd ones. I don’t think the non(super)commutative algebra $$\mathbf C_{\mathrm s} =\mathbf R\oplus i\Pi\mathbf R$$ has anything to do with the former.