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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.

MSC:
14M15 Grassmannians, Schubert varieties, flag manifolds
14P25 Topology of real algebraic varieties
14A20 Generalizations (algebraic spaces, stacks)
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[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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