Tetraplectic structures compatible with local quaternionic toric actions

TL;DR

This paper develops a quaternionic analogue of toric geometry, introducing local quaternionic torus actions, invariants for classification, and quaternionic Lagrangian fibrations, culminating in a Delzant-type classification of orbit spaces.

Contribution

It introduces the theory of local quaternionic torus actions, invariants for their classification, and quaternionic Lagrangian fibrations, extending toric geometry into the quaternionic setting.

Findings

Defined characteristic pair and Euler class invariants for quaternionic torus actions.

Established a quaternionic version of the Arnold-Liouville theorem for Lagrangian fibrations.

Classified orbit spaces as quaternionic integral affine manifolds with corners.

Abstract

This paper introduces a quaternionic analogue of toric geometry by developing the theory of local $Q^n := Sp(1)^n$-actions on 4n-dimensional manifolds, modeled on the regular representation. We identify obstructions that measure the failure of local properties to globalize and define two invariants: a combinatorial invariant called the characteristic pair and a cohomological invariant called the Euler class, which together classify local quaternionic torus actions up to homeomorphism. We also study tetraplectic structures in quaternionic toric geometry by introducing locally generalized Lagrangian-type toric fibrations and show that such fibrations are locally modeled on $\mathbb{R}^n\times Q^n$ using a quaternionic version of the Arnold-Liouville theorem. In the last part, we show that orbit spaces of these actions acquire the structure of quaternionic integral affine manifolds with corners and Lagrangian overlaps, and we classify such spaces by establishing a quaternionic Delzant-type theorem.