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Kähler-Ricci flow on homogeneous toric bundles

Assume that $X$ is a homogeneous toric bundle of the form $G^{\mathbb{C}}\times_{P,τ} F$ and is Fano, where $G$ is a compact semisimple Lie group with complexification $G^\mathbb{C}$, $P$ a parabolic subgroup of $G^\mathbb{C}$, $τ:P\rightarrow (T^m)^\mathbb{C}$ 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 Kähler-Ricci flow on $X$ with a $G\times T^m$-invariant initial Kähler form in $c_1(X)$ converges, modulo the algebraic torus action, to a Kähler-Ricci soliton. This extends a previous work of X. H. Zhu. As a consequence we recover a result of Podestà-Spiro.