Higher dimensional residues in complex analysis

While it's hard to guess exactly what you are after, I believe such residues were first studied by Poincaré (Sur les résidus des intégrales doubles, Acta Math. 9 (1887) 331–380), with your sought invariance encoded in the statement that "residue" takes a closed p-form on the complement of a hypersurface to a closed (p–1)-form on the hypersurface, so that in cohomology it induces a map $\smash{\mathrm{Res}:H^p(X\setminus S)\to H^{p-1}(S)}$. Modern references are, for instance:

• P. A. Griffiths, Poincaré and algebraic geometry, Bull. Amer. Math. Soc. 6 (1982) 147–159.

• E. Cattani and A. Dickenstein, Introduction to residues and resultants, pp. 1–61 in Solving polynomial equations, Springer-Verlag, Berlin (2005) (PDF).

• A. Yger, The concept of "residue” after Poincaré: cutting across all of mathematics, pp. 225–241 in The scientific legacy of Poincaré, Amer. Math. Soc. (2010) (French PDF).