Kähler-Ricci flow on homogeneous toric bundles

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 $G^{\mathbb{C}}$, $\tau :P\to (T^m{)}^{\mathbb{C}}$ 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.

to appear in International Journal of Mathematics

References in corpus (2)

• KAWA lecture notes on the Kähler-Ricci flow
• Kähler-Ricci flow on a toric manifold with positive first Chern class

Cited by in corpus (3)

• Kähler-Ricci solitons on horospherical variety
• The moduli space of Fano manifolds with Kähler-Ricci solitons
• Kähler-Ricci flow on horospherical manifold