On the conditional measures on the orbits of the complex torus

TL;DR

This paper investigates the structure of invariant measures on compact Kähler manifolds with Hamiltonian torus actions, deriving formulas for conditional measures on orbits and applying these to the uniqueness of solutions to the g-Monge-Ampère equation.

Contribution

It introduces a formula for conditional measures on complex torus orbits and applies it to establish a uniqueness result for the g-Monge-Ampère equation solutions.

Findings

Derived explicit formulas for conditional measures on torus orbits.

Proved a conditional uniqueness statement for the g-Monge-Ampère equation.

Enhanced understanding of measure structure in Hamiltonian torus actions.

Abstract

We explore the structure of invariant measures on compact K\"ahler manifolds with Hamiltonian torus actions. We derive the formula for conditional measures on the orbits of the complex torus and use it to prove a conditional statement about uniqueness of solutions to the $g$-Monge-Amp\`ere equation.