Consider three functions $f, g$ and $h$ on a smooth curve $X$ over $\mathbb{C}$.
I have found the following equality:
$$\sum (res(f\frac{dg}{g})\frac{dh}{h}-res(f\frac{dh}{h})\frac{dg}{g})=0.$$
Here the sum is over complex points of $X$, $res$ means residue of the corresponding meromorphic form on $X$. The most important point is that object
$$res(f\frac{dg}{g})\frac{dh}{h}$$ should be considered as an element of the module of absolute Kahler differentials $\Omega^1_{\mathbb{C}/\mathbb{Q}}.$
For $f=1$ it gives the Weil reciprocity law, since $$res(\frac{dg}{g})\frac{dh}{h}-res(\frac{dh}{h})\frac{dg}{g}=ord(g)\frac{dh}{h}-ord(h)\frac{dg}{g}=$$
$$=\frac{d\{g,h\}_W}{\{g,h\}_W},$$
where $\{g,h\}_W$ stands for the Weil symbol. So,
$$0=\sum (res(\frac{dg}{g})\frac{dh}{h}-res(\frac{dh}{h})\frac{dg}{g})=
=\sum\frac{d\{g,h\}_W}{\{g,h\}_W}=\frac{d\prod \{g,h\}_W}{\prod \{g,h\}_W}.$$
From this it follows that $\prod \{g,h\}_W$ is algebraic complex number. Since the product of Weil symbols depends continously on the functions, it should be constant. But for $g=h=1$ it is equal to one. This finishes the proof of the Weil reciprocity law.
Question 1: How one can prove this formula? I have found quite a long proof with Newton polygons, following ideas of Askold Khovansky.
Question 2: Can one interprete it as some kind of a reciprocity law for surfaces?
Question 3: I am almost sure that it is known. I would be very grateful for any reference.
Best Answer
Look at so-called Parshin reciprocity law, for example Mazin' approach http://www.math.utoronto.ca/mmazin/thesis.pdf
I guess that is exactly what you want.
If I am not mistaken, your version follows from the standard version of Weil reciprocity law for the functions $f/g,f/h$, in the spirit of logarithmic differentials (cf. http://www.math.toronto.edu/askold/osaka.pdf) If not, it is very strange.