Ciprian S. Borcea (Rider University)

ABSTRACT: The possible configurations, up to orientation-preserving

isometry, for a planar n-gon with prescribed length for each of

its edges, make-up a compact space, which is, in general, a smooth,

orientable manifold of dimension (n-3). Its topological type varies

according a chamber structure for admissible edge-length-vectors,

and can be investigated by means of Morse theory, geometric invariant

theory, symplectic and toric geometry.

In adequate coordinates, the defining equations are algebraic,

and yield families of complex projective varieties whose real

points are the above configuration spaces. In particular, a

construction used by Darboux for quadrilaterals, leads, in arbitrary

dimension, to Calabi-Yau varieties. The singularities of the

latter are away from the real locus, and resolutions to Calabi-Yau

manifolds will contain identifiable types of special Lagrangians.

These resolved Darboux varieties can be presented as codimension two

complete intersections in the toric variety associated to a

permutohedron.

A conjecture of Strominger, Yau, and Zaslow suggests that

mirror symmetry for pairs of Calabi-Yau manifolds corresponds

geometrically to a duality of fibrations in special Lagrangian

tori, and indeed, we do find special Lagrangian tori at appropriate

points in our families.

We also detect the mirror families (using Batyrev-Borisov duality)

and retrieve the polygon spaces as special Lagrangians on the

mirror Calabi-Yau manifolds.

These phenomena are related to the root system A_{n-1}, and there

are similar developments for the BC_n type. This is enough for

propagating our examples to a significantly larger class of

Calabi-Yau manifolds which arise in connection with certain

reflexive centrally symmetric polytopes.

A different, yet related complexification, and thus other examples

of special Lagrangian tori on Calabi-Yau manifolds, can be obtained

from the non-Euclidean scenario. Up to birational transformations,

the Euclidean case appears as a limit of the non-Euclidean case.