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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)
##### Keywords:
Grassmannian; polylogarithms; real polygon
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##### References:
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