The \(k\)th polylogarithm \(\ln_ k x\), \(| x | < 1\), is defined as \(\ln_ k x = \sum_{n = 1}^ \infty {x^ n \over n^ k}\), \(k=1,2, \dots\). One sees that, for \(k \geq 2\) and \(| x | < 1\), \(\ln_ k x = \int_ 0^ x \ln_{k - 1} z\) \({dz \over z}\), and thus \(\ln_ k x\) can be analytically continued to a multi-valued holomorphic function on \(\mathbb{C} \backslash \{0,1\}\). These polylogarithms and suitable expressions in terms of them are expected to play the role of higher regulators from the algebraic \(K\)-theory of complex algebraic varieties to their Deligne cohomology. Precise conjectures for the \(K\)-theory of (number) fields were stated by Zagier and Goncharov. A rather detailed account of the Bloch-Wigner dilogarithm \(D_ 2 (x) = \ln_ 2 x + \log | x | \arg (1 - x)\) is given and it is shown that this \(D_ 2\) can be used to define a map \(K_ 3 (\mathbb{C}) \to H^ 1_{\mathcal D} (\text{Spec} (\mathbb{C}), \mathbb{R}(2)) = \mathbb{C}/ \mathbb{R}(2)\) which is just the Beilinson regulator (or, equivalently, half of Borel’s regulator). Also, the Beilinson-Deligne construction of the regulator \(c_ 2 : K_ 2 (X) \to H^ 2_{\mathcal D} (X, \mathbb{Z} (2))\) for a curve \(X\), involving the dilogarithm, is presented in detail. This construction goes back to Bloch in case of an elliptic curve. Here the approaches of \(H^ 2_{\mathcal D} (X, \mathbb{Z} (2))\) in terms of line bundles and connections on the curve \(X\), due to Deligne and Ramakrishnan, are recalled.
One of the aspects of polylogarithms discussed from several points of view is the (variation of) mixed Hodge structure associated to them. First, studying the monodromy of the \(\ln_ k\), one is led to define the polylogarithm local system. Actually, this is the inverse limit of local systems, the so-called \(n\)th polylogarithm local systems. It is shown that the \(n\)th polylogarithm local system on a smooth complex algebraic curve \(X\) (with smooth compactification \(\overline X)\) underlies a good variation of mixed Hodge structure whose weight graded quotients are canonically isomorphic to the Tate structures \(\mathbb{Q}=\mathbb{Q} (0)\), \(\mathbb{Q} (1), \dots, \mathbb{Q}(n)\). Also, for a point \(P \in D=\overline X \backslash X\) and a tangent vector \(\vec v\) at \(P\), one has the limit mixed Hodge structure associated to \(\vec v\). For \(X = \mathbb{C} \backslash \{0,1\}\) and \(P = 0\) or \(P = 1\), \(\vec v = \partial/ \partial z\) or \(\vec v = - \partial/\partial z\), respectively, the limit mixed Hodge structures can be given explicitly. They involve the powers up to \(n\) of \((2 \pi i)\) and, at \(P = 1\), the values \(\zeta (2),\dots, \zeta (n)\) of the Riemann zeta function. As a matter of fact, the value of \((k - 1)! \ln_ k1\) is \((k-1)! \zeta(k)\) modulo \(\mathbb{Z} (k) = (2 \pi i)^ k \mathbb{Z}\). For a smooth variety \(X\) over \(\mathbb{C}\) such that \(H^ 1 (X)\) is pure of weight 2 a mixed Hodge structure on the fundamental group \(\pi_ 1 (X,x)\) can be defined. In fact, the (completed) group algebra \(\mathbb{Q} \pi_ 1(X,x)^ \wedge\) carries a canonical mixed Hodge structure. For its construction one uses a result of Chen giving a canonical isomorphism between \(\mathbb{C} \pi_ 1 (X,x)^ \wedge\) and a complete Hopf algebra \(A^ \wedge\) constructed by means of differential forms. This result is proved using iterated integrals, whose definition and basic properties are briefly recalled.
By the way, as a matter of fact the polylogarithms \(\ln_ kx\) can also be written as suitable iterated integrals. This mixed Hodge structure is then related to a unipotent variation of mixed Hodge structure. For a smooth variety \(X\) the notion of a Tate variation of mixed Hodge structure is introduced. An example of such a variation is the one defined by the polylogarithms. It is shown that for a smooth variety \(X\) with smooth compactification \(\overline X\) and normal crossings divisor \(D = \overline X \backslash X\) the canonical extension of a Tate variation of mixed Hodge structure to \(\overline X\) is trivial as a holomorphic vector bundle. Coming back to polylogarithms and regulators and writing \({\mathcal H}\) for the category of mixed Hodge structures, one has a canonical isomorphism \(\text{Ext}^ 1_{\mathcal H} (\Lambda, \Lambda (m)) \simeq \mathbb{C}/ \Lambda (m)\), \(\Lambda = \mathbb{Z}, \mathbb{Q}\) or \(\mathbb{R}\). Thus the regulator \(K_ m (\mathbb{C}) \to \mathbb{C}/ \Lambda (m)\) can be interpreted as a map \(K_ m (\mathbb{C}) \to \text{Ext}^ 1_{\mathcal H} (\Lambda, \Lambda (m))\). For \(c_ 2 : K_ 2 (X) \to H^ 2_{\mathcal D} (X,\mathbb{Z} (2))\) as above one can also interprete the construction in terms of mixed Hodge structures. For the \(n\)th polylogarithm local system one shows that as a variation of mixed Hodge structure it is an extension of \(\mathbb{Q}\) by the shift \((\text{Sym}^{n - 1} V_ x) \otimes \mathbb{Q}(1)\), where \(V_ x \in \text{Ext}^ 1_{\mathcal H} (\mathbb{Q}, \mathbb{Q} (1))\) is the variation of mixed Hodge structure over \(\mathbb{C} \backslash \{0,1\}\) corresponding to the function \(x \in {\mathcal O}^{\{0,1\}}\).
The paper closes with a motivic description of the polylogarithm variation which goes back to Deligne. It gives the polylogarithm variation as a quotient of a variation of mixed Hodge structure over \(\mathbb{C} \backslash \{0,1\}\) with fiber over \(\vec v\) given by a certain quotient of \(\mathbb{Q} \pi_ 1 (\mathbb{C} \backslash \{0,1\}, \vec v)^ \wedge\).
For the entire collection see [Zbl 0788.00054].