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The tame symbol. (Le symbole modéré.) (French) Zbl 0749.14011
Let \(X\) be a complex analytic variety with holomorphic structure sheaf \(\mathcal O\). Also, let \(G\) be a commutative complex analytic group. \(G(\mathcal O)\) will denote the sheaf of holomorphic \(G\)-valued functions on \(X\) and \(G(\mathbb C)\) is the constant sheaf \(G\), i.e., the sheaf of locally constant functions on \(X\). In particular, for \(G=\mathbb{G}_ m\), one has \(G(\mathcal O)=\mathcal O^*\), the sheaf of invertible holomorphic functions on \(X\).
Starting point for the paper is the construction of a morphism in the derived category of sheaves of abelian groups on \(X\), \[ \mathcal O^\ast \overset{\mathbb{L}}\otimes G(\mathcal O)\longrightarrow [G(\mathcal O)\longrightarrow\Omega^ 1\otimes\mathrm{Lie}(G)]_{-1,0}(1),\tag{1} \] where the subscript \(-1,0\) means that the complex is placed in degrees \(-1\) and \(0\), and where (1) denotes the Tate twist. Using the quasi-isomorphism \([\mathbb{Z}(1)\longrightarrow\mathcal O]_{- 1,0}\overset{\sim}{\longrightarrow} \mathcal O^\ast\), this amounts to define \[ [{\mathbb{Z}}(1)\longrightarrow{\mathcal O}]_{-1,0}\otimes G({\mathcal O})\longrightarrow [G({\mathcal O})\longrightarrow\text{Lie}(G)\otimes\Omega^ 1]_{-1,0}(1).\tag{2} \] For \({\mathbb{Z}}(1)\otimes G({\mathcal O})\longrightarrow G({\mathcal O})(1)\) one takes the identity, and after a choice of \(i\in{\mathbb{C}}\), the map \({\mathcal O}\otimes G({\mathcal O})\longrightarrow\text{Lie}(G)\otimes\Omega^ 1\cong\text{Lie}(G)\otimes\Omega^ 1(1)\) can be given by \(f\otimes g\mapsto\frac{1}{2\pi i}f\cdot g^{-1}dg\). For the cohomology of (1) one obtains morphisms \[ H^ i(X,{\mathcal O}^*)\otimes H^ j(X,G({\mathcal O}))\longrightarrow {\mathbb{H}}^{i+j}(X, [G({\mathcal O})\longrightarrow\Omega^ 1\otimes\text{Lie}(G)]_{- 1,0}(1)).\tag{3} \] In particular, for \(i=j=0\), one gets an isomorphism class of holomorphic \(G(1)\)–torsors with connection \(\nabla\) associated to elements \(f\in{\mathcal O}^*\) and \(g\in G({\mathcal O})\). Write \((f,g]\) for the representative with auxiliary data given by the logarithms of \(f\), i.e., a choice of \(\log f\) gives a trivialisation \(\{\log f,g\}\) of \((f,g]\). One has \(\nabla\{\log f,g\}=\frac{1}{2\pi i}\log f\cdot g^{-1}dg\), \(\{\log f+n\cdot 2\pi i,g\}=\{\log f,g\}\cdot g^ n\), and for the curvature of \((f,g]\) one finds \(R=\frac{1}{2\pi i}\cdot\frac{df}{f}\wedge g^{- 1}dg\in\Omega^ 2\otimes\text{Lie}(G)\).
Assume \(X\) is a Riemann surface \(\Sigma\). For \(x\in\Sigma\), \(f\) invertible on \(\Sigma-\{x\}\) and \(g:\Sigma-\{x\}\longrightarrow G\), one may combine (2) with the residue map in \(x\) to obtain \((f,g]_x\in G(\mathbb C)\). After a choice of \(i\in \mathbb C\), this gives just the monodromy \((f,g]_ S\) of \((f,g]\) on an oriented circle \(S\) around \(x\). Special cases include (i) \(G= \mathbb G_ a\) and (ii) \(G= \mathbb G_m\). In (i) one finds \((f,g]_ x=\frac{1}{2\pi i}\oint \frac{df}{f}g=\mathrm{Res}_x\left(\frac{df}{f}\cdot g\right)\); in (ii) \((f,g]_x\), also traditionally written \((f,g)_x\), becomes the tame symbol: \((f,g)_x= (-1)^{v(f)v(g)} \left[g^{v(f)}/f^{v(g)}\right](x)\), where \(v\) denotes the valuation at \(x\). For \(G=\mathbb G_m\) the product \(\mathcal O^\ast\overset{\mathbb L}{\otimes}\mathcal O^\ast\rightarrow [\mathcal O^\ast\xrightarrow{df/f}\Omega^1](1)\) defines the product in the Beilinson-Deligne cohomology for \(\mathbb Z(1)_{\mathcal D}\otimes\mathbb Z(1)_{\mathcal D}\longrightarrow\mathbb Z(2)_{\mathcal D}\), thus leading to an interpretation of the \(\mathcal O^\ast(1)\) – (or \(\mathcal O/(2\pi i)^2{\mathbb Z})\) – torsor with connection \((f,g]\), written \((f,g)\) in this situation, in terms of (variations of) mixed Hodge structures. This torsor \((f,g)\) is the opposite of the one of holomorphic families on X of iterated mixed Hodge extensions of \(\mathbb Z(0)\) by \(\mathbb Z(1)\), by \(\mathbb Z(2)\), inducing extensions \([f]\) of \(\mathbb Z(0)\) by \(\mathbb Z(1)\) and \([g](1)\) of \(\mathbb Z(1)\) by \(\mathbb Z(2)\). Such a variation of mixed Hodge structures defines a horizontal section of \((f,g)\) and conversely. A nice example is provided by \(X=\mathbb P^1(\mathbb C)\setminus\{0,1,\infty\}\), \(f=1-z\) and \(g=z\). A trivialising horizontal section of \((1-z,z)\) is given by \(\{\log(1- z),z\}+\mathrm{Li}_ 2(z)\), where \(\mathrm{Li}_ 2(z)\) is the dilogarithm function. A choice of \(\log(1-z)\) determines \(\mathrm{Li}_ 2(z)\) up to \((2\pi i)^ 2\mathbb Z\).
Furthermore, the symbol \((f,g)\) has bimultiplicativity and symmetry properties corresponding to symbols in \(K\)-theory, e.g., \((f,g)+(g,f)\) is trivialised by \(\{\log f,g\}+\{\log g,f\}-\log f\log g\), which, in the example of \(\mathbb P^1(\mathbb C)\setminus\{0,1,\infty\}\) is reflected by the relation for the dilogarithm: \[ \mathrm{Li}_2(z)+\mathrm{Li}_2(1-z)+\log(z)\log(1-z)=\zeta(2)=-{(2\pi i)^ 2/24}. \] The foregoing construction of \((f,g)\) can be extended in the following way: Let \(\Lambda_ 1\) and \(\Lambda_ 2\) be local systems of finitely generated free \(\mathbb Z\)-modules and let \(B:\Lambda_ 1\otimes\Lambda_ 2\longrightarrow\mathbb Z\) be a bilinear form. This gives rise to a morphism \((\Lambda_ 1\otimes\mathcal O^\ast)\overset{\mathbb L}{\otimes}(\Lambda_ 2\otimes\mathcal O^\ast)\longrightarrow\mathcal O^\ast\overset{\mathbb L}{\otimes}\mathcal O^\ast\), which, combined with (1) for \(G=\mathbb G_m\), leads to \[ (\Lambda_1\otimes\mathcal O^\ast)\overset{\mathbb L}{\otimes}(\Lambda_2\otimes\mathcal O^\ast)\longrightarrow [\mathcal O^\ast\longrightarrow\Omega^1](1).\tag{4} \] For sections \(f\) of \(\Lambda_ 1\otimes\mathcal O^\ast\) and \(g\) of \(\Lambda_ 2\otimes\mathcal O^\ast\) one obtains a \(\mathcal O^(1)\)- - (or \(\mathcal O/(2\pi i)^ 2\mathbb Z\)–)torsor with connection, again written \((f,g)\), and a choice of \(\log f\in\Lambda_ 1\otimes\mathcal O\) determines a section \(\{\log f,g\}_ B\) of \((f,g)\) with properties similar to \(\{\log f,g\}\) of \((f,g)\) above.
Again, with a choice of \(i\in\mathbb C\) and an oriented circle \(S\), one may calculate the monodromy \((f,g)_ S\) of \((f,g)\) on \(S\). Let \(T:\tilde{S}\longrightarrow\tilde{S}\), where \(\tilde{S}\) is a universal covering of \(S\), denote the monodromy of \(\tilde{S}\). Then, for locally constant global sections \(f\) of \(\Lambda_ 1\otimes\mathcal O^*\) and \(g\) of \(\Lambda_ 2\otimes\mathcal O^*\), one finds the formula: \((f,g)_ S=\exp\left(2\pi iB((T-1)\log f/2\pi i,\log g/2\pi i)\right)\). Moreover, if \(\Lambda_ 1\) and \(\Lambda_ 2\) are dual, and \(T\) does not have the eigenvalue \(1\), the locally constant global sections of \(\Lambda_ i\otimes\mathcal O^*\) on \(S\) form a finite group (\(i=1,2\)), and the symbol \((f,g)_ S\) defines the Pontrjagin duality between these groups.
One may go a step further and weaken the regularity properties of \(f\) and \(g\) on the circle \(S\). It is shown that one can take the Sobolev spaces \(\mathcal O_ s\) and \(\mathcal O_ t\), \(s+t\geq 1\), and a local system \(\Lambda\) of finitely generated \(\mathbb Z\)–modules on \(S\), with dual \(\Lambda^{\vee}\), to get a pairing \((\mathcal O^*_ s\otimes\Lambda)(S)\otimes(\mathcal O^*_ t\otimes\Lambda^{\vee})(S)\longrightarrow\mathbb C^*\). In particular, for \(s+t=1\), the pairing \((f,g)_ S\) extends to a pairing \[ (\mathcal O^*_ s\otimes\Lambda)(S)\otimes(\mathcal O^*_ t\otimes\Lambda^{\vee})(S)\longrightarrow\mathbb C^*, \] making both factors Cartier duals of each other. The case \(s=\frac{1}{2}\) is to be used in the construction of the Heisenberg group defined by the central extension of \((\mathcal O^*_{1/2}\otimes\Lambda)(S)\) with \(\mathbb C^*\). One needs some preliminary results. First, let \(\Sigma^ 0\) be the interior of the Riemann surface \(\Sigma\) with (non–empty) boundary \(\partial\Sigma\) and let \(j:\Sigma^ 0\hookrightarrow\Sigma\) the inclusion. Write \(\mathcal O_ s\), \(s\in\mathbb R\), for the sheaf on \(\Sigma\), equal to the subsheaf of \(j_ *\mathcal O\) consisting of holomorphic functions with values in \(\mathcal O_ s\) at \(\partial\Sigma\) (with connected components the circles \(S_{\alpha}\) (\(\alpha\in J\))). In general, the Sobolev space \(\mathcal O_ s\) on the boundary \(\partial\Sigma\), i.e. on the \(S_{\alpha}\), is not equal to the restriction of the latter \(\mathcal O_ s\) to \(\partial\Sigma\). Furthermore, one defines \(\mathcal O^*_ s\) by the exact exponential sequence \(0\longrightarrow 2\pi i\mathbb Z\longrightarrow\mathcal O_ s\longrightarrow\mathcal O^*_ s\longrightarrow 0\), and also on the boundary, \(\mathcal O^*_ s=\mathcal O_ s/2\pi i\mathbb Z\). One proves that, in case \(\Sigma\) is compact, the section \(f\in H^ 0(\partial\Sigma, \mathcal O^*_ s\otimes\Lambda)\) is the restriction to \(\partial\Sigma\) of a section of \(\mathcal O^*_ s\otimes\Lambda\) on \(\Sigma\) if and only if for any section \(u\) of \(\mathcal O^*\otimes\Lambda^{\vee}\) on \(\Sigma\), holomorphic up to the boundary, one has \((f,u)_{\partial\Sigma}= \prod_{\alpha}(f,u)_{S_{\alpha}}=1\).
Take \(s=\frac{1}{2}\) and let \(B\) be a positive definite bilinear form on the local system \(\Lambda\) on \(\Sigma\). Then, for \(f\in H^ 0(\partial\Sigma,\mathcal O^*_{1/2}\otimes\Lambda)\), \(B\) induces an inner product \(B( , )_{\partial\Sigma}\) and one shows that, if \(f\) is the boundary value of a holomorphic section \(u\) of \(\mathcal O^*\otimes\Lambda\) on \(\Sigma^ 0\), and \(\bar f\) the complex conjugate of \(f\), then \((*)\) \(B(f,\overline{f})_{\partial\Sigma}\geq 1\), with equality if and only if \(u\) is locally constant. This follows from an explicit expression for the curvature of the line bundle with connection \((u,\overline{u})\) on the two dimensional \(\text{C}^{\infty}\)-manifold (with boundary) \(\Sigma\).
For compact \(\Sigma\) with boundary \(\partial\Sigma\), let \(\text{L}\) be the sheaf \(\Lambda\otimes\mathcal O^*_{1/2}\) on \(\Sigma\), and use the same notation for the sheaf \(\Lambda\otimes\mathcal O^*_{1/2}\) on \(\partial\Sigma\). Thus the restriction of \(\text{L}(\Sigma)\) to \(\partial\Sigma\) embeds into \(\text{L}(\partial\Sigma)\), but is not surjective! Also, assume an even symmetric bilinear form \(B:\Lambda\otimes\Lambda\longrightarrow\mathbb Z\) and a central extension \(E\) of \(\Lambda\) by \(\mathcal O^*_{\infty}\) are given. The commutator \((x,y)=xyx^{-1}y^{-1}:E\times E\longrightarrow E\) factors over \(( , ):\Lambda\times\Lambda\longrightarrow\mathcal O^*_{\infty}\). Suppose that for two local sections \(\lambda\), \(\mu\) of \(\Lambda\) one has: \((\lambda,\mu)=(-1)^{B(\lambda,\mu)}\). One wants to construct a central extension with splitting over \(\text{L}(\Sigma)\), i.e. a Heisenberg group, of the following form:
\[ \begin{tikzcd} 1 \ar[r] & \mathbb{C}^\ast \ar[r] & \mathrm{L}(\partial\Sigma)^{\sim} \ar[r] & \mathrm{L}(\partial\Sigma) \ar[r] & {}\\ &&& \mathrm L(\Sigma) \ar[ul]\ar[u, "\mathrm{restriction}"'] & \end{tikzcd}\quad.\tag{5} \]
It is enough to restrict to a connected component (oriented circle) \(S\) of \(\partial\Sigma\). Let \(\text{L}\) be a sheaf of groups on \(S\) and let \(A\) be an abelian group (constant sheaf). One defines a central extension of \(\text{L}\) by the stack of \(A\)–torsors as the data of a gerb \({\mathcal G}(h)\) bound by \(A\), for each local section \(h\) of \(\text{L}\), with natural compatibility properties. With two sections \(h_ 1\), \(h_ 2\) of \(\text{L}\) one associates the commutator \((h_ 1,h_ 2)_{\mathcal G}\), an object of \({\mathcal G}((h_ 1,h_ 2))\).
For a global section \(h\), the gerb \(\mathcal G(h)\) defines an \(A\)-torsor \(\displaystyle\int_ S{\mathcal G}(h)\): the set of isomorphism classes of global objects of \(\mathcal G(h)\). This construction defines a central extension of \(\mathrm{L}(S)\) by \(A\), written \(\displaystyle\int_ S{\mathcal G}\), in Grothendieck’s interpretation. The commutator defined by \(\displaystyle{\int_ S{\mathcal G}}\), \(\displaystyle{\text{L}(S)\times\text{L}(S)\longrightarrow\int_ S{\mathcal G}}\), associates with sections \(h_ 1\), \(h_ 2\) of \(\text{L}\) the isomorphism class of \((h_ 1,h_ 2)_{\mathcal G}\) in \({\mathcal G}((h_ 1,h_ 2))\).
For the actual construction of \({\mathcal G}\) on \(\Sigma\) and \(\partial\Sigma\) a sheaf of auxiliary data \({\mathcal K}\) to define a trivialisation of \({\mathcal G}(h)\), is introduced. Here the bilinear form \(B\) enters. The final result for the commutators is independent of \({\mathcal K}\): (i) \((h_ 1,h_ 2)_{\mathcal G}=B(h_ 1,h_ 2)\); (ii) for the central extension of \(\text{L}(S)\) by \({\mathbb{C}}^*\), \((h_ 1,h_ 2)=B(h_ 1,h_ 2)_ S\). For positive definite \(B\), \(h_ 1\in\text{L}(\Sigma)\) and \(h_ 2=\overline{h}_ 1\), one obtains by \((*)\), \((h_ 1,\overline{h}_ 1)\geq 1\).

MSC:
14F99 (Co)homology theory in algebraic geometry
14H55 Riemann surfaces; Weierstrass points; gap sequences
14F20 Étale and other Grothendieck topologies and (co)homologies
14L30 Group actions on varieties or schemes (quotients)
30F99 Riemann surfaces
14D07 Variation of Hodge structures (algebro-geometric aspects)
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
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References:
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