The K\"ahler-Ricci flow on projective bundles
  We study the behaviour of the K\"ahler-Ricci flow on projective bundles. We show that if the initial metric is in a suitable K\"ahler class, then the fibers collapse in finite time and the metrics converge subsequentially in the Gromov-Hausdorff sense to a metric on the base. 
Jian Song, G\'abor Sz\'ekelyhidi, Ben Weinkove
math.DG