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Geometry of configurations, polylogarithms, and motivic cohomology. (English) Zbl 0863.19004
This paper gives a full proof account of the state of the art of the theory of polylogarithms from a geometric point of view as advocated by the author. The case of the dilogarithm being known since some time by work of Gabrielov et al., Bloch, Wigner, Zagier,\( \dots\), the underlying paper focusses on the trilogarithm.
Classically the \(p\)-th polylogarithm \(\text{Li}_p (z)\) is defined as the analytic continuation of the expression \(\text{Li}_p(z)=\sum_{n=1}^\infty {z^n \over n^p}\), \(|z|\leq 1\). With \(\text{Li}_1(z)= - \log(1-z)\), one has the inductive formula \(\text{Li}_p(z)=\int^z_0 \text{Li}_{p-1} (z) {dt \over t}\) with its (multivalued) continuation to \(\mathbb{P}^1_\mathbb{C} \backslash \{0,1, \infty\}\). It turns out to be advantageous to consider modified functions \({\mathcal L}_p (z)={\mathcal R}_p \left( \sum^p_{j=0} {2^j B_j \over j!} (\log |z |)^j \cdot \text{Li}_{p-j} (z) \right)\), where the \(B_j\) are the Bernoulli numbers, and \({\mathcal R}_m\) denotes the real part for odd \(m\) and the imaginary part for even \(m\), \(\text{Li}_0 (z):= -{1 \over 2}\). Thus one has \[ {\mathcal L}_2 (z)={\mathfrak I} \bigl( \text{Li}_2 (z) \bigr) + \arg (1-z) \cdot \log |z |, \] the Bloch-Wigner function. For the present paper the most interesting function becomes \[ {\mathcal L}_3 (z)={\mathfrak R} \Bigl( \text{Li}_3 (z)-\log |z |\cdot \text{Li}_2 (z)+ \textstyle {{1\over 3}} \log^2 |z|\cdot \text{Li}_1(z) \Bigr). \] The functions \({\mathcal L}_p(z)\) are single-valued, real analytic on \(\mathbb{P}^1_\mathbb{C} \backslash \{0,1, \infty\}\) and continuous at \(0,1,\infty\). In particular, \({\mathcal L}_3 (0)= {\mathcal L}_3 (\infty) =0\) and \({\mathcal L}_3 (1)= \zeta_\mathbb{Q} (3)\), the Riemann zeta-function at \(s=3\). The functions \({\mathcal L}_p (z)\) admit a Hodge theoretic interpretation.
As a first goal one tries to find the generic functional equation for the (modified) trilogarithm. Motivated by results on the Bloch-Wigner function where the functional equations of the dilogarithm are related to the cross-ratio of four points in \(\mathbb{P}^1\), in the case of the trilogarithm one considers configurations of six (or seven) points in \(\mathbb{P}^2\). By an explicit geometric reasoning the main result is obtained: The generic functional equation for the trilogarithm \({\mathcal M}_3\) (a specific alternating sum of \({\mathcal L}_3\)’s) is a seven term identity for \({\mathcal M}_3\) on a configuration \((l_0, \dots, l_6)\) of seven points in \(\mathbb{P}^2\) that can be given explicitly in terms of the coordinates of the points. In particular, the Spence-Kummer relation may be derived. Furthermore, it is shown that the trilogarithm is determined by its functional equation.
Polylogarithms show up in other contexts, e.g. in connection with algebraic \(K\)-theory, motivic cohomology, characteristic classes, continuous cohomology, the Dedekind zeta function of an arbitrary number field, \(\dots\), etc. In some cases one can prove interesting results, e.g. for a number field \(F\) one can express (up to a non-zero rational factor) the value of \(\zeta_F (2)\) in terms of the dilogarithm \({\mathcal L}_2\) at specific values of its argument depending on the (complex) embeddings of \(F\). A similar result holds for \(\zeta_F (3) \) in terms of \({\mathcal L}_3\). For general \(\zeta_F (n)\), \(n=4,\dots\), Zagier stated the conjecture that they can be expressed, analogously to \(\zeta_F (2)\) and \(\zeta_F (3)\), in terms of \({\mathcal L}_n\). This fits very well in Beilinson’s world where values of \(L\)-functions at special values of their arguments are given (up to non-zero rational factors) by the volume of the regulator map which is itself a map from algebraic \(K\)-groups to Deligne-Beilinson cohomology. As a matter of fact, the theory of polylogarithms is closely related to \(K\)-theory. One of its main building blocks is a certain complex \(\Gamma_F (n)\) (the existence of which was originally conjectured by Beilinson and Lichtenbaum) of the form: \[ \Gamma_F (n): {\mathcal B}_n(F) @> \delta>> {\mathcal B}_{n-1} (F) \otimes F^\times @>\delta>> \cdots @>\delta>> {\mathcal B}_2 (F) \otimes \wedge^{n-2} F^\times @>\delta>> \wedge^n F^\times, \] where \({\mathcal B}_m (F) = \mathbb{Z} [\mathbb{P}^1 _F]/{\mathcal R}_m (F)\), with \({\mathcal R}_m (F) \subset \mathbb{Z} [\mathbb{P}^1_F]\) reflecting the functional equations of the classical \(m\)-polylogarithm. Here \({\mathcal B}_n (F)\) is placed in degree one, and the \(\delta\)’s are explicitly defined. For the \(K\)-groups of \(F\) one has \(K_n (F )_\mathbb{Q} = \text{Prim} H_n (GL_n(F), \mathbb{Q})\) and by the canonical filtration on \(H_n (GL_n (F), \mathbb{Q})\) implied by \(\text{Im} (H_n (GL_{n-1} (F), \mathbb{Q}) \to H_n (GL_n(F), \mathbb{Q}))\), one obtains a filtration \(K_n(F)_\mathbb{Q} \supset K_n^{(1)} (F)_\mathbb{Q} \supset K_n^{(2)} (F)_\mathbb{Q} \supset \cdots \). Let \(K_n^{[i]} (F)_\mathbb{Q}: = K_n^{(i)} (F)_\mathbb{Q}/K_n^{(i+1)} (F)_\mathbb{Q}\). Then one has Conjecture A: \(K_{2n-i}^{[n-i]} (F)_\mathbb{Q} = H^i (\Gamma_F(n) \otimes \mathbb{Q})\). On the other hand, Beilinson conjectured the existence of a mixed Tate category \({\mathcal M}_T(F)\) which should be Tannakian. Thus the formalism of Tannakian categories implies that \({\mathcal M}_T (F)\) is equivalent to the category of finite-dimensional representations of some graded pro-Lie algebra \(L(F)_\bullet = \oplus^{-\infty}_{i= -1} L(F)_i\). One may state Conjecture B: (i) \(L(F)_{\leq-2}\) is a free graded pro-Lie algebra such that the dual of the space of its degree \(-n\) generators is isomorphic to \({\mathcal B}_n (F)_\mathbb{Q}\); (ii) The dual map to the action of the quotient \(L(F)_\bullet /L(F)_{\leq -2}\) on the space of degree \(-(n-1)\) generators of \(L(F)_{\leq -2}\) is just the differential \(\delta: {\mathcal B}_n (F)_\mathbb{Q} \to({\mathcal B}_{n-1} (F)\otimes F^\times)_\mathbb{Q}\). It is shown that in Beilinson’s world Conjecture A is equivalent to Conjecture B. Conjecture B has some deep consequences, e.g. its truth implies the truth of a conjecture of Bogomolov, and also of a conjecture due to Shafarevich which says that the commutant of \(\text{Gal} (\overline \mathbb{Q}/ \mathbb{Q})\) is a free profinite group.
Let \(F\) be an arbitrary field and define \(B_p(F): = \mathbb{Z} [\mathbb{P}^1_F \backslash \{0,1, \infty\}]/R_p(F)\), \(p\leq 3\), where the \(R_p(F)\) again reflect the functional equations of the classical \(p\)-th polylogarithm. One defines the complex \(B_3(F) \otimes \mathbb{Q}\) as follows: \(B_3(F)_\mathbb{Q} @>\delta>> (B_2(F) \otimes F^\times)_\mathbb{Q} @>\delta>> (\wedge^3 F^\times)_\mathbb{Q}\), with \(B_3 (F)_\mathbb{Q}\) placed in degree 1, and \(\delta \{x\} = [x] \otimes x\) and \(\delta ([x] \otimes y) = (1-x) \wedge x \wedge y\) for a generator \(\{x\}\) of \(B_3(F)\) and a generator \([x]\) of \(B_2(F)\). Then there are canonical maps \(c_1: K_5^{[2]} (F)_\mathbb{Q} \to H^1 (B_3(F) \otimes \mathbb{Q})\) and \(c_2: K_4^{[1]} (F)_\mathbb{Q} \to H^2 (B_3(F) \otimes \mathbb{Q})\). It is conjectured that \(c_1\) and \(c_2\) are isomorphisms. This should be related to results of Suslin on Milnor \(K\)-groups.
Other subjects discussed are duality of configurations, projective duality, and an explicit formula for the Grassmannian trilogarithm.
Many unsolved questions and deep conjectures remain.

MSC:
19F27 Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects)
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
14F20 Étale and other Grothendieck topologies and (co)homologies
11R42 Zeta functions and \(L\)-functions of number fields
11R70 \(K\)-theory of global fields
33E20 Other functions defined by series and integrals
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