Geometric construction of polylogarithms.

*(English)*Zbl 0824.14043It 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.

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.

Reviewer: A.N.Parshin (Moskva)

##### MSC:

14M15 | Grassmannians, Schubert varieties, flag manifolds |

14P25 | Topology of real algebraic varieties |

14A20 | Generalizations (algebraic spaces, stacks) |

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\textit{M. Hanamura} and \textit{R. MacPherson}, Duke Math. J. 70, No. 3, 481--516 (1993; Zbl 0824.14043)

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##### References:

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[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 |

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[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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