residue calculation in the definition of the regulator on $K_2$

Let $C$ be a smooth projective curve defined over $\mathbb{Q}$. The regulator is a map $$\operatorname{reg}: K_2(C)_{\mathbb{Q}} \longrightarrow H^1(C(\mathbb{C}), \mathbb{R}).$$ Here $K_2(C)_{\mathbb{Q}}$ is the rational second K-group which admits the description:
$$\ker\left(K_2(\mathbb{Q}(C)) \longrightarrow \bigoplus_{x \in C(\overline{\mathbb{Q}})} \overline{\mathbb{Q}}^\times\right) \otimes_{\mathbb{Z}} \mathbb{Q}$$ where $K_2(\mathbb{Q}(C))$ is generated by symbols $\{f, g\}$ and the map is given by the tame symbols $$T_x(\{f, g\})=(-1)^{ord_x(f)ord_x(g)} [f^{ord_x g}/g^{ord_x f}](x).$$

Since $H^1(C(\mathbb{C}), \mathbb{Q})$ is the dual of $H_1(C(\mathbb{C}), \mathbb{Q})$, to define the regulator of a symbol $\{f, g\}$, one needs to attach a real number $$\operatorname{reg}(\{f, g\})(\gamma)$$ to each $\gamma \in H_1(C(\mathbb{C}), \mathbb{Q})$.

To do so, one first starts defining $$\omega(f, g)=\log |f|\operatorname{d} \arg(g)-\log |g| \operatorname{d} \arg(f)$$ which is a closed real analytic 1-form on the complement of $S=\operatorname{div}(f) \cup \operatorname{div}(g)$ on the Riemann surface $C(\mathbb{C})$. We first define the real number $$\frac{1}{2\pi i} \int_{\gamma} \omega(f, g)$$ for $\gamma$ a cycle on $U=X-S$ (and notice that it actually only depends on the cohomology class of $\gamma$). Then we would like to say that this definition remains valid also for $\gamma$ in the whole of $X$. Since the cohomologies of $U$ and $X$ are related by residue maps $$0 \to H^1(X) \to H^1(U) \to \bigoplus_{s \in S} \mathbb{R} \to \mathbb{R} \to 0$$ one needs to show that the residue vanishes at all points of $S$. What I have read is that this residue is given by $$\log | T_s(\{f, g\})|$$ (so since the tame symbol is torsion equal to zero) but I don't understand how to prove this. I guess I should take a small circle $C_s$ around a point in $S$ and show that $$\frac{1}{2\pi} \int_{C_s} \omega(f, g)=\log |T_s(\{f, g\})|.$$ Can anybody help me proving this? This should be some kind of Cauchy theorem but since the integrand is not holomorphic I am a bit lost...

This may no longer be relevant, but I thought it would be instructive to pin down the mistakes in the question which lead to the problem with the residue computation. The main problem originates in the interpretation of the target of the regulator (leading to mixing of two possible interpretations). The relevant instance of the Beilinson regulator for the question should be a map $$K_2(C)\to H^{2}_D(C,\mathbb{Q}(2)),$$ from K-theory to Deligne-Beilinson cohomology. The more refined analytic information contained in Deligne-cohomology is used in both definitions of the regulator described below (so it would appear to not be possible to define the regulator only with the information contained in singular cohomology, as in the question).

The first approach where one has to compute a residue to extend the cohomology class from an open part to the whole curve is the one outlined in

• S. Bloch. The dilogarithm and extensions of Lie algebras. In: Algebraic K-theory, Evanston 1980, Lecture Notes in Math. 854, Springer, 1981, pp. 1-23.

An alternative outline of the approach along with more information on Deligne-Beilinson cohomology can be found in Section 1 of

• H. Esnault and E. Viehweg. Deligne-Beilinson cohomology. In: Beilinson's conjectures on special values of L-functions. Perspect. Math. 4, Academic Press, 1988, pp. 43-91.

This approach is based on an alternative description of the relevant Deligne cohomology group $H^2_D(C,\mathbb{R}(2))$ via holomorphic line bundles with connections, classified by $H^1(C,\mathbb{C}^\times)$. Bloch's paper contains a construction of a holomorphic line bundle with connection over $\mathbb{C}^\times\times\mathbb{C}^\times$, based on the dilogarithm. For rational functions $f,g$, we get a map $U=C\setminus(\operatorname{div}(f)\cup\operatorname{div}(g))\to\mathbb{C}^\times\times\mathbb{C}^\times$ along which we can pull back the line bundle. Mapping a symbol to the class of the line bundle with connection produces a proto-regulator map $K_2(k(C))\to\lim_{S\subset C} H^1(C\setminus S,\mathbb{C}^\times)$ where $S$ runs through the finite subsets of the curve $C$ (because we have to remove the divisors for the rational functions in the symbol).

The formulas in the proof Proposition 1.10 in Bloch's paper (or the explicit cup-product formula in 1.4.iv) of the Esnault-Viehweg paper), giving an explicit description of the regulator, can be interpreted as the proper refinement of the formula for $\omega(f,g)$ in the question (the second problem in the question is that one should not take the logarithm of the absolute value, but construct a cocycle for a covering such that the logarithm is defined on each of the opens; this way everything stays nicely holomorphic).

Now we want to extend this to the whole curve, by showing the vanishing of a residue map in a localization sequence for Deligne cohomology (to compare to the localization sequence for K-theory); that map is actually $\exp(2\pi i\operatorname{res}(-))$ (the third problem is that the boundary map is not exactly the residue; the exponential arises from the definition of the Deligne complex). The computation of the residue map applied to a symbol is done in Proposition 1.19 of Bloch's paper, it is given by the tame symbol (as opposed to the logarithm of the tame symbol in the question). To show this, one reduces to very specific symbols $(f,t)$ (with $t$ a local parameter at the point $x$ we are looking at) where the form defining the holomorphic line bundle (locally around $x$) is $\frac{1}{2\pi i}\log f\frac{\operatorname{d}t}{t}$; applying the residue map for Deligne cohomology mentioned above and a very simple Cauchy application finish the job. Then the regulator extends to $K_2(C)\to H^2_D(C,\mathbb{R}(2))$: elements in $K_2(C)$ have trivial tame symbol and therefore the associated Deligne cohomology classes have trivial residue.

There is a second approach where the regulator can be interpreted as somehow related to $H^1(C,\mathbb{Q})$. For an open subcurve $U$, one considers $H^1_D(U,\mathbb{R}(1))\subset H^1(U,\mathbb{C})$ which are (closed modulo exact) $C^\infty$-forms invariant under complex conjugation. Next, $H^1_D(C,\mathbb{R}(1))$ is actually a direct summand of $H^1_D(C,\mathbb{R}(1))$. One can define the regulator associated to $(f,g)$ by taking forms $\alpha,\beta\in H^1_D(U,\mathbb{R}(1)$ having divisors $\operatorname{div}(f)$ and $\operatorname{div}(g)$, respectively, taking their cup-product and then project down to $H^1_D(C,\mathbb{R}(1))$. This is all explained in detail in Section 1 of

• C. Deninger and K. Wingberg. On the Beilinson conjectures for elliptic curves with complex multiplication. In: Beilinson's conjectures on special values of L-functions. Perspect. Math. 4, Academic Press, 1988, pp. 249-272.

Tracing through the definitions of cup product and projection to $H^1_D(C,\mathbb{R}(1))$ provides another way of rectifying the definition of $\omega(f,g)$ in the question (and one doesn't pair with a homology cycle on the curve but with a real holomorphic 1-form). This approach doesn't actually compute residues but instead uses a well-defined projection $H^1_D(U,\mathbb{R}(1))\to H^1_D(C,\mathbb{R}(1))$ (using the refined analytic information in Deligne cohomology as opposed to singular cohomology).