I am studying the on-shell diagrams techniques to compute scattering amplitudes, let's take as a common reference this paper: http://arxiv.org/abs/1212.5605.
Trying to put the content of the article in a formal way I have stumbled upon the problem of giving rigor to this operation:
Suppose you have a complex manifold $M$ of dimension $n$ and a codimension one submanifold $N$. Actually this is an analitic surface defined as the zero loci of an holomorphic function $s$. In addition suppose that it is possible to write locally $M = N \times \mathbb{C}$. On $M$ we are given a $n$-dimensional meromorphic form $\omega$, such that $ord_N \ \omega = 1$. I guess that this should allow to find, close to $N$, appropriate coordinates $(z,c)$ that implements the above factorization of $M$ (i.e. (z,0) is a generic point of $N$ whilst (z,c) is a generic point in $M$ close to $N$) and such that
$\omega = h(z,c)/c \wedge_{i=1}^{n-1} dz_i \wedge dc,$
with $h \ne 0$.
One then could define an operation that produce a $n-1$ form on $N$ by mean of an "integration over the $c$ variable". By this I mean that one can define on $N$ the form $\omega' = (\oint h(z,c)/c dc) \wedge_{i=1}^{n-1} dz_i = h(z,0) \wedge_{i=1}^{n-1} dz_i$. This operation could also be thought as "taking the residue of $\omega$ over $N$".
This kind of integrations are used often in the paper, when studying "singularities" of the amplitudes as "residues" of the grassmannian differential form $d\alpha/\alpha \ \delta(C \cdot Z)$
Do you know any reference where I could look for something similar to the above?