Local torus actions modeled on the standard representation

TL;DR

This paper introduces the concept of local torus actions modeled on the standard representation, providing topological invariants for classification and exploring their relation to locally toric Lagrangian fibrations.

Contribution

It defines invariants for local torus actions, classifies them topologically, and connects these actions to locally toric Lagrangian fibrations with new criteria.

Findings

Local torus actions are classified by a characteristic pair and an Euler class.

A topological classification of locally standard torus actions generalizes previous results.

Conditions are provided for when a local torus action corresponds to a locally toric Lagrangian fibration.

Abstract

We introduce the notion of a local torus action modeled on the standard representation (for simplicity, we call it a local torus action). It is a generalization of a locally standard torus action and also an underlying structure of a locally toric Lagrangian fibration. For a local torus action, we define two invariants called a characteristic pair and an Euler class of the orbit map, and prove that local torus actions are classified topologically by them. As a corollary, we obtain a topological classification of locally standard torus actions, which is a generalization of the topological classification of quasi-toric manifolds by Davis and Januszkiewicz and of effective two-dimensional torus actions on four-dimensional manifolds without nontrivial finite stabilizers by Orlik and Raymond. We investigate locally toric Lagrangian fibrations from the viewpoint of local torus actions. We give a necessary and sufficient condition in order that a local torus action becomes a locally toric Lagrangian fibration. Locally toric Lagrangian fibrations are classified by Boucetta and Molino up to fiber-preserving symplectomorphisms. We shall reprove the classification theorem of locally toric Lagrangian fibrations by refining the proof of the classification theorem of local torus actions. We also investigate the topology of a manifold equipped with a local torus action when the Euler class of the orbit map vanishes.