Higher dimensional residues in complex analysis

Question

Consider a function $f:\mathbb{CP}^1\times\mathbb{CP}^1\to \mathbb{CP}^1 $ defined by $f([x_1,x_2],[y_1,y_2])=[x_1y_1,x_2y_2]$. This function is well defined except at $([0,1],[1,0])$ or vice versa (in therms of the Riemann sphere $\mathbb{C}_\infty$ we do not have a well defined zero times infinity). Now consider a function $g:U\to\mathbb{C}$ analytic in in a neighbourhood $U$ of $([0,1],[1,0])$ in $\mathbb{CP}^1\times\mathbb{CP}^1$. I am looking for a way to define some sort of numerical value of residue to the product $fg$ at the point $([0,1],[1,0])$.

I am looking for any possible definition of `residue' which make sense and is invariant to change of coordinates. This may be in terms of integral on a polydisk (feel free to introduce a Kahler form if it makes things easier). It may be in terms of cohomology (as long as it is calculable and not completely abstract). Anything will do.

The motivation lies in the theory of integrable systems, and there there is a rather unsatisfactory answer - define a standard example as you wish, and then use the huge number of symmetries of the integrable system to define the other cases which can arise. Of course this may do a particular job, but in general is not a good way to proceed, so I am looking for a more sensible theoretical definition. (I am obviously not an expert in algebraic geometry, all I can say is that from a physical point of view, such a construction seems possible.)

Answer 1

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).

Answer 2

I am not an expert in the field, however I believe the Gohberg-Sigal theory on constructing the residues (and defining the ranks) of meromorphic operator-functions could be of great help:

An operator generalization of the logarithmic residue theorem and the theorem of Rouché ,I. Ts. Gokhberg, E. I. Sigal, Mat. Sb. (N.S.), 1971, Volume 84(126), Number 4, Pages 607–629 (Mi msb3169).