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} 
Frederick Tsz-Ho Fong
math.DG math.CV