Lagrangian skeleta, collars and duality

TL;DR

This paper explores a geometric duality connecting skeleta in cotangent bundles of projective spaces with collars of local surfaces, revealing deep links between symplectic and toric dualities.

Contribution

It provides a new geometric realization of duality between skeleta and collars, integrating symplectic and toric dualities into a unified framework.

Findings

Established a correspondence between Lagrangian submanifolds and vector bundles.

Described birational transformations dual to vector bundle deformations.

Linked symplectic duality with toric duality through geometric constructions.

Abstract

We present a geometric realization of the duality between skeleta in $T^*\mathbb P^n$ and collars of local surfaces. Such duality is predicted by combining two auxiliary types of duality: on one side, symplectic duality between $T^*\mathbb P^n$ 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.