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Geometric construction of polylogarithms. (English) Zbl 0824.14043
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 I. M. Gelfand and 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.

14M15 Grassmannians, Schubert varieties, flag manifolds
14P25 Topology of real algebraic varieties
14A20 Generalizations (algebraic spaces, stacks)
Full Text: DOI
[1] A. A. Beilinson, A. B. Goncharov, V. V. Schechtman, and A. N. Varchenko, Aomoto dilogarithms, mixed Hodge structures and motivic cohomology of pairs of triangles on the plane , The Grothendieck Festschrift, Vol. I, Progr. Math., vol. 86, Birkhäuser Boston, Boston, MA, 1990, pp. 135-172. · Zbl 0737.14003
[2] A. Beĭ linson, R. MacPherson, and V. Schechtman, Notes on motivic cohomology , Duke Math. J. 54 (1987), no. 2, 679-710. · Zbl 0632.14010 · doi:10.1215/S0012-7094-87-05430-5
[3] I. M. Gelfand and R. D. MacPherson, Geometry in Grassmannians and a generalization of the dilogarithm , Adv. in Math. 44 (1982), no. 3, 279-312. · Zbl 0504.57021 · doi:10.1016/0001-8708(82)90040-8
[4] A. Goncharov, Geometry of configurations, polylogarithms and motivic cohomology , · Zbl 0863.19004 · doi:10.1006/aima.1995.1045
[5] R. Hain and R. MacPherson, Higher logarithms , Illinois J. Math. 34 (1990), no. 2, 392-475. · Zbl 0737.14014
[6] M. Hanamura and R. MacPherson, in preparation.
[7] Lewin, L., ed., Structural properties of polylogarithms , Mathematical Surveys and Monographs, vol. 37, American Mathematical Society, Providence, RI, 1991. · Zbl 0745.33009
[8] J. Yang, The Hain-MacPherson trilogarithm, the Borel regulators and the value of the Dedekind zeta function at \(3\) ,
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