# K\"ahler-Ricci flow on homogeneous toric bundles

## TL;DR

This paper proves that the normalized Kähler-Ricci flow on certain homogeneous toric bundles converges to a Kähler-Ricci soliton, extending previous results and recovering known theorems.

## Contribution

It demonstrates convergence of the Kähler-Ricci flow on homogeneous toric bundles to solitons, generalizing prior work by Zhu and recovering results by Podestà-Spirò.

## Key findings

- Flow converges to a Kähler-Ricci soliton
- Extension of Zhu's work to broader class of bundles
- Recovers a known result of Podestà-Spirò

## Abstract

Assume that $X$ is a homogeneous toric bundle of the form $G^{\mathbb{C}}\times_{P,\tau} F$ and is Fano, where $G$ is a compact semisimple Lie group with complexification $G^\mathbb{C}$, $P$ a parabolic subgroup of $G^\mathbb{C}$, $\tau:P\rightarrow (T^m)^\mathbb{C}$ is a surjective homomorphism from $P$ to the algebraic torus $(T^m)^\mathbb{C}$, and $F$ is a compact toric manifold of complex dimension $m$. In this note we show that the normalized K\"{a}hler-Ricci flow on $X$ with a $G\times T^m$-invariant initial K\"{a}hler form in $c_1(X)$ converges, modulo the algebraic torus action, to a K\"{a}hler-Ricci soliton. This extends a previous work of X. H. Zhu. As a consequence we recover a result of Podest\`{a}-Spiro.