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The symplectic geometry of polygons in Euclidean space. (English) Zbl 0889.58017
Let $$M_r$$ be the moduli space of $$n$$-gons with fixed side lengths $$(r_1, \dots, r_n)$$ in Euclidean space. The space $$M_r$$ is naturally homeomorphic to $$\widetilde M_r/SO (3)$$, where $$\widetilde M_r =\{u\in (S^2)^n: \sum r_ju_j =0\}$$.
The authors give $$M_r$$ a symplectic and a complex analytic structure by the following philosophy: whenever a compact Lie group $$G$$ and its complexification $$G^\mathbb{C}$$ operate on a complex manifold $$M$$ with a given $$G$$-invariant Kähler form, one obtains a $$G$$-equivariant moment map $$\mu$$ and $$\mu^{-1} (0)/G \approx G^\mathbb{C} \mu^{-1} (0)/G^\mathbb{C}$$; the left hand side is a “symplectic space” and the right hand side a “complex analytic space”. By beautiful and natural geometric constructions, the authors produce functions which Poisson-commute and make a dense open subset $$M_r'$$ of $$M_r$$ into a toric variety. They also identify functions which give action-angle coordinates on a dense open subset $$M_0$$ of $$M_r'$$.
There is some overlap with the work of A. Klyachko [Aspects Math. E 25, 67-84 (1992; Zbl 0820.51016)].