On dimension reduction in the K\"ahler-Ricci flow

We consider dimension reduction for solutions of the K"ahler-Ricci flow with nonegative bisectional curvature. When the complex dimension $n=2$, we prove an optimal dimension reduction theorem for complete translating K"ahler-Ricci solitons with nonnegative bisectional curvature. We also prove a general dimension reduction theorem for complete ancient solutions of the K"ahler-Ricci flow with nonnegative bisectional curvature on noncompact complex manifolds under a finiteness assumption on the Chern number $c_1^n$.