Geometric construction of polylogarithms
It is well known that some cycles on Grassmannian manifolds have a close connection with analytic functions called polylogarithms (they generalize the classical dilogarithm). This was discovered by \textit{I. M. Gelfand} and \textit{R. D. MacPherson} [Adv. Math. 44, 279-312 (1982; Zbl 0504.57021)]. Later several new constructions have appeared. The paper under review is one of them. The authors consider an arbitrary real polygon \(P\) in an Euclidean space. Fix a complex projective space \(\mathbb{P}^ n\). A \(P\)-figure is an assignment of a linear subspace of \(\mathbb{P}^ n\) to any face of \(P\) with reasonable properties. The authors show how to attach to any \(P\)-figure a cocycle on a Grassmannian represented by a differential form. The integrals of these forms will be (roughly speaking) the desired polylogarithms. The paper is written very nicely. The author explains the trivial case of \(n=1\) (usual logarithm), then more the complicated case of \(n = 2\) (dilogarithm) and concentrate their efforts on the really difficult case of \(n = 3\). They hope to extend their construction to arbitrary \(n\) in the future.
scientific article; zbMATH DE number 558974
scientific article