# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] C. E. Contou-Carrère,Jacobienne locale, groupe de bivecteurs de Witt universel et symbole local modéré, à paraître. [2] J. Giraud,Cohomologie non abélienne, Grundlehren,179, Springer Verlag, 1971. [3] J.-P. Serre,Groupes algébriques et corps de classes, Paris, Hermann, 1959. · Zbl 0097.35604 [4] Séminaire de géométrie algébrique du Bois-Marie, dirigé parA. Grothendieck. [5] Théoire des topos et cohomologie étale des schémas, t. III :Lecture Notes in Mathematics,305, Springer Verlag, 1973. [6] Groupes de monodromie en géométrie algébrique, t. I :Lecture Notes in Mathematics,288, Springer Verlag, 1972.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.