On non-geometric augmentations in high dimensions

TL;DR

This paper constructs high-dimensional Legendrian submanifold augmentations that are not derived from Lagrangian fillings, revealing new algebraic obstructions and exploring their relation to augmentation varieties and spherical spuns.

Contribution

It introduces non-geometric augmentations in high dimensions and analyzes their obstructions, expanding understanding beyond traditional geometric origins.

Findings

Existence of non-geometric augmentations in high dimensions.

Obstructions identified via Seidel's isomorphism and algebraic map injectivity.

Relation between augmentation varieties and spherical spuns discussed.

Abstract

In this note we construct augmentations of Chekanov-Eliashberg algebras of certain high dimensional Legendrian submanifolds that are not induced by exact Lagrangian fillings. The obstructions to the existence of exact Lagrangian fillings that we use are Seidel's isomorphism and the injectivity of a certain algebraic map between the corresponding augmentation varieties proven by Gao and Rutherford. In addition, along the way we discuss the relation between augmentation varieties of Legendrian submanifolds and their spherical spuns.