I was reading about [Calabi-Yau manifolds](http://en.wikipedia.org/wiki/Calabi-Yau_manifold), about which I know little, and was wondering
if these (or related complex manifolds, perhaps [K3 surfaces](http://en.wikipedia.org/wiki/K3_manifold)) can be viewed as configuration
spaces (or [moduli spaces](http://en.wikipedia.org/wiki/Moduli_space)) 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!



Calabi-Yau manifolds and polygonal linkage configuration spaces: related?
An illustrative example: *the moduli space $M$ of regular pentagons with edges of unit length*. This embeds as an open, dense subset of a compact complex surface $\bar{M}$ with a canonical Kaehler form. This surface, a 4-fold blow-up of $\mathbb{CP}^2$, is not Calabi-Yau (trivial canonical bundle) but Fano (ample anticanonical bundle).


The compactified regular pentagon space $\bar{M}$ is the space of 5-tuples of unit vectors in $\mathbb{R}^3$ with centre of mass zero, modulo the diagonal action of $SO(3)$. Since we remember the order of these vectors, typical points represent regular pentagons with a distinguished vertex ("start here") and adjacent edge ("go this way"). There are also points which represent an equilateral triangle together with a pair of antipodal points, and these non-pentagon points form ten 2-spheres in $\bar{M}$. 


$\bar{M}$ has a natural symplectic structure, for which the ten 2-spheres are Lagrangian. Take the unique area-form on $S^2$, invariant under $SO(3)$, of total area 1 and inducing the complex orientation of $S^2=\mathbb{CP}^1$. The moment map $S^2\to \mathfrak{so}(3)^\ast \cong \mathbb{R}^3$ for the $SO(3)$-action is just the inclusion of $S^2$ into $\mathbb{R}^3$. The product $(S^2)^5$ carries the product symplectic form, again $SO(3)$-invariant, with moment map $\mu(x\_1,\dots,x\_5)=x\_1+\dots + x\_5\in \mathbb{R}^3$. The symplectic quotient $\mu^{-1}(0)/SO(3)$ is just $\bar{M}$.


The action of $SO(3)=PU(2)$ respects the complex structure of $(\mathbb{CP}^1)^5$, and $\bar{M}$ inherits a complex structure by Kaehler reduction. It turns out to be isomorphic as a Kaehler surface to a blow up of $\mathbb{CP}^2$ at four special points with a Fano Kaehler form (but I haven't thought through which points). See Seidel's [Lectures on 4-dimensional Dehn twists](http://arxiv.org/abs/math/0309012), ex. 1.10. There's a natural action of the icosahedral group, permuting the $x\_i$.


If one wanted pentagons defined by some other linear equation, say $a\_1x\_1+\dots + a\_5x\_5=0$, one would give the $S^2$-factors areas $a\_i$. 


One can also interpret $\bar{M}$ as an algebro-geometric (GIT) quotient of $(\mathbb{CP}^1)^5$ by $PSL\_2(\mathbb{C})$. The quotient happens to be the Deligne-Mumford (or Grothendieck-Knudsen) compactification $\bar{M}\_{0,5}(\mathbb{C})$ of configurations of five points on $\mathbb{CP}^1$. The real points $\bar{M}\_{0,5}(\mathbb{R})$, the fixed points of an anti-holomorphic involution of $\bar{M}$, are also interesting: their connected components are polyhedral and are copies of the 2-dimensional [Stasheff associahedron](http://en.wikipedia.org/wiki/Associahedron) (a.k.a. pentagon).


**References:** 


F. Kirwan, [Cohomology of quotients in symplectic and algebraic geometry](http://books.google.com/books?id=4wfZBnlSaJ0C&printsec=frontcover&dq=kirwan+cohomology+of+quotients&source=bl&ots=WhidvP7PUy&sig=4VbvWVflmsmYZ0F7PrdpO4-G1hg&hl=en&ei=C_aATYuYBovmsQP_g_SSBg&sa=X&oi=book_result&ct=result&resnum=2&ved=0CCAQ6AEwAQ#v=onepage&q&f=false).


J.-C. Hausmann and A. Knutson, [Polygon spaces and Grassmannians](http://arxiv.org/pdf/dg-ga/9602012v1).


