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.)