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Lagrangian skeleta, collars and duality

Edoardo Ballico, Elizabeth Gasparim, Francisco Rubilar, Bruno Suzuki

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.

Lagrangian skeleta, collars and duality

Abstract

We present a geometric realization of the duality between skeleta in and collars of local surfaces. Such duality is predicted by combining two auxiliary types of duality: on one side, symplectic duality between and a crepant resolution of the 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.
Paper Structure (9 sections, 7 theorems, 64 equations, 4 figures)

This paper contains 9 sections, 7 theorems, 64 equations, 4 figures.

Table of Contents

  1. Skeleton to collar duality
  2. Lagrangian skeleton of $T^*\mathbb P^n$
  3. Example: skeleton of $T^*\mathbb P^3$
  4. General case: the skeleton of $T^*\mathbb P^n$.
  5. Potentials on the cotangent bundle
  6. Birational maps within the skeleton
  7. Duality for multiplicity $n$ singularities
  8. Vector bundles on local surfaces
  9. Deformations

Key Result

Theorem 1.1

The following diagram commutes:

Figures (4)

  • Figure 1: $Z_n$ as toric resolution of $\mathcal{X}_n$
  • Figure 2: $\widetilde{Y_{n}}$ as toric resolution of $\mathcal{X}^\vee_n=Y_n$
  • Figure : Duality between Lagrangians and vector bundles.
  • Figure :

Theorems & Definitions (10)

  • Theorem 1.1
  • Proposition 3.1
  • Proposition 6.2
  • Corollary 6.3
  • Lemma 6.5
  • proof
  • Proposition 6.6
  • proof
  • Proposition 6.7
  • proof