On the collapsing rate of the K\"ahler-Ricci flow with finite-time singularity

This short note studies the collapsing behavior of the K"ahler-Ricci flow on a compact K"ahler manifold X admitting a holomorphic submersion X -> B where B is a K"ahler manifold of lower dimension than X. We give cohomological and curvature conditions under which the fibers collapse at the optimal rate ~(T-t)^{1/2}