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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!

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up vote 7 down vote accepted

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, 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 (a.k.a. pentagon).

References:

F. Kirwan, Cohomology of quotients in symplectic and algebraic geometry.

J.-C. Hausmann and A. Knutson, Polygon spaces and Grassmannians.

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Thank you for this erudite and instructive reply! I've studied the space of unit pentagons through a nice article in the American Mathematical Monthly by Don Shimamoto and Catherine Vanderwaart ("Spaces of polygons in the plane and Morse theory." Volume 112, pages 289-310, 2005). But I was unaware of the connection to complex manifolds. Thanks! –  Joseph O'Rourke Mar 16 '11 at 11:42
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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.

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Thanks for the reference---It seems directly apropos. –  Joseph O'Rourke Mar 16 '11 at 11:45
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