Kähler–Ricci flow on homogeneous toric bundles

Abstract

Assume that [Formula: see text] is a homogeneous toric bundle of the form [Formula: see text] and is Fano, where [Formula: see text] is a compact semisimple Lie group with complexification [Formula: see text], [Formula: see text] a parabolic subgroup of [Formula: see text], [Formula: see text] is a surjective homomorphism from [Formula: see text] to the algebraic torus [Formula: see text], and [Formula: see text] is a compact toric manifold of complex dimension [Formula: see text]. In this note, we show that the normalized Kähler–Ricci flow on [Formula: see text] with a [Formula: see text]-invariant initial Kähler form in [Formula: see text] converges, modulo the algebraic torus action, to a Kähler–Ricci soliton. This extends a previous work of Zhu. As a consequence, we recover a result of Podestà–Spiro.