TITLE: Calabi-Yau manifolds and polygonal linkage configuration spaces: related?
QUESTION [9 upvotes]: I was reading about Calabi-Yau manifolds, about which I know little, and was wondering
if these (or related complex manifolds, perhaps K3 surfaces) can be viewed as configuration
spaces (or moduli spaces) of articulated polygonal linkages, say with fixed edge lengths and
vertex joints?  This question is a shot in the dark; apologies for its naiveté.  But it might help
me understand complex manifolds if I can view them as polygon configuration spaces.
Any references in this general direction would be appreciated.  Thanks!

REPLY [6 votes]: Tim's example has been generalised somewhat by Kapovich and Millson. The space $\mathcal{M}(\mathbf{r})$ of $n$-gons in $\mathbb{E}^3$ with a fixed vector $\mathbf{r}\in\mathbb{R}_+^n$ of side lengths carries the structure of a complex-analytic space with at worst quadratic singularities. This is because $\mathcal{M}(\mathbf{r})$ is a weighted symplectic quotient of the Kähler manifold $(S^2)^n$ by $SO(3)$.
Reference: Michael Kapovich and John J. Millson, The symplectic geometry of polygons in Euclidean space. J. Differential Geom. Volume 44, Number 3 (1996), 479-513.