Kähler-Ricci flow and conformal submersion

We study singularity formation of Kähler-Ricci flow on a Kähler manifold that admits a horizontally homothetic conformal submersion into another Kähler manifold. We will derive necessary and sufficient conditions for the preservation of horizontally homothetic conformal submersion along the flow and establish the formation of type I singularity together with a standard splitting of the Cheeger-Gromov limit. This generalizes the setup of Calabi symmetry that was discussed in \cite{1} and \cite{2}, thus gives new proofs for the results listed there.