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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 $$c_1: K_5^{} (F)_\mathbb{Q} \to H^1 (B_3(F) \otimes \mathbb{Q})$$ and $$c_2: K_4^{} (F)_\mathbb{Q} \to H^2 (B_3(F) \otimes \mathbb{Q})$$. 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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