A Note on Wu-Zheng's Splitting Conjecture

TL;DR

This paper proves a relation between Ricci rank and splitting dimension in Kähler-Ricci flows, confirming a conjecture by Wu-Zheng under bounded curvature conditions, and refines Cao's splitting theorem.

Contribution

It establishes that the splitting dimension equals the complex dimension minus the Ricci rank, confirming Wu-Zheng's conjecture for bounded curvature cases.

Findings

Splitting dimension equals n minus Ricci rank.

Confirms Wu-Zheng's splitting conjecture under bounded curvature.

Refines Cao's splitting theorem with Ricci rank relation.

Abstract

Cao's splitting theorem says that for any complete K\"ahler-Ricci flow $(M,g(t))$ with $t\in [0,T)$, $M$ simply connected and nonnegative bounded holomorphic bisectional curvature, $(M,g(t))$ is holomorphically isometric to $\C^k\times (N,h(t))$ where $(N,h(t))$ is a Kahler-Ricci flow with positive Ricci curvature for $t>0$. In this article, we show that $k=n-r$ where $r$ is the Ricci rank of the initial metric. As a corollary, we also confirm a splitting conjecture of Wu-Zheng when curvature is assumed to be bounded.