Geometric construction of polylogarithms
type: Item
instance of: scholarly article
title: Geometric construction of polylogarithms
zbMATH Open document ID: 0824.14043
DOI: 10.1215/S0012-7094-93-07010-X
author: Masaki Hanamura
author: Robert MacPherson
published in: Duke Mathematical Journal
publication date: 1995-11-13T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 14M15
Mathematics Subject Classification ID: 14P25
Mathematics Subject Classification ID: 14A20
zbMATH DE Number: 558974
zbMATH Keywords: Grassmannian
zbMATH Keywords: polylogarithms
zbMATH Keywords: real polygon
reviewed by: A. N. Parshin
MaRDI profile type: Publication
cites work: Q3972721
cites work: Notes on motivic cohomology
cites work: Geometry in Grassmannians and a generalization of the dilogarithm
cites work: Geometry of configurations, polylogarithms, and motivic cohomology
cites work: Higher logarithms
cites work: Q4001952
full work available at URL: s0012-7094-93-07010-x
OpenAlex ID: W4256590942
Recommended article: Q4873729
Recommended article: Functional equations of iterated integrals with regular singularities
Recommended article: The cycle complex over \(\mathbb{P}^1\) minus 3 points: toward multiple zeta value cycles
Recommended article: Grassmannian trilogarithms.
Recommended article: Q5483282
Recommended article: Double filtration of twisted logarithmic complex and Gauss-Manin connection
Recommended article: The Jacobian module, the Milnor fiber, and the \(D\)-module generated by \(f^s\)
Recommended article: Q3736056
Recommended article: A simple construction of Grassmannian polylogarithms
Recommended article: Identities involving (doubly) symmetric polynomials and integrals over Grassmannians
description: scientific article; zbMATH DE number 558974