Scalar Curvature and Stability of Toric Fibrations

TL;DR

This paper investigates the stability of toric fibrations over flag varieties by constructing test configurations, computing Futaki invariants, and analyzing the Mabuchi functional to understand geometric stability.

Contribution

It introduces a method to analyze stability of toric fibrations using symplectic data and generalized Pick's Theorem, providing new tools for geometric analysis.

Findings

Constructed test configurations of toric fibrations

Computed Futaki invariants using symplectic data

Derived a simplified form of the Mabuchi Functional

Abstract

We study fibrations $\cV$ of toric varieties over the flag variety $G/T$, where $G$ is a compact semisimple Lie group and $T$ is a maximal torus. From symplectic data, we construct test configurations of $\cV$ and compute their Futaki invariants by employing a generalization of Pick's Theorem. We also give a simple form of the Mabuchi Functional.