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.