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.
Categories: math.DG math.AP