# Reduced phase space and toric variety coordinatizations of Delzant spaces

## TL;DR

This paper describes explicit coordinate systems for Delzant spaces from symplectic and algebraic geometry perspectives, providing formulas and proofs that clarify their structure and relations.

## Contribution

It introduces explicit coordinate transformations for Delzant spaces as reduced phase spaces and toric varieties, enhancing understanding of their geometric structures.

## Key findings

- Explicit formulas for coordinate transformations between reduced phase space and toric variety
- Identification of maximal coordinate neighborhoods around fixed points
- Simplified proofs of fundamental properties of Delzant spaces

## Abstract

In this note we describe the natural coordinatizations of a Delzant space defined as a reduced phase space (symplectic geometry view-point) and give explicit formulas for the coordinate transformations. For each fixed point of the torus action on the Delzant polytope, we have a maximal coordinatization of an open cell in the Delzant space which contains the fixed point. This cell is equal to the domain of definition of one of the natural coordinatizations of the Delzant space as a toric variety (complex algebraic geometry view-point), and we give an explicit formula for the toric variety coordinates in terms of the reduced phase space coordinates. We use considerations in the maximal coordinate neighborhoods to give simple proofs of some of the basic facts about the Delzant space, as a reduced phase space, and as a toric variety. These can be viewed as a first application of the coordinatizations, and serve to make the presentation more self-contained.