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.

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