Lagrangian skeleta, collars and duality

Abstract

We present a geometric realization of the duality between skeleta in $T^{\ast }{\mathbb{P}}^n$ and collars of local surfaces. Such duality is predicted by combining two auxiliary types of duality: on one side, symplectic duality between $T^{\ast }{\mathbb{P}}^n$ and a crepant resolution of the $A_n$ singularity; on the other side, toric duality between two types of isolated quotient singularities. We give a correspondence between Lagrangian submanifolds of the cotangent bundle and vector bundles on collars, and describe those birational transformations within the skeleton which are dual to deformations of vector bundles.

arXiv commentTo appear in Springer Proceedings in Mathematics & Statistics. Vol. Birational Geometry, Kahler - Einstein Metrics, and Degenerations