The continuity method on Fano fibrations

Abstract

We study finite-time collapsing limits of the continuity method. When the continuity method starting from a rational initial K\"ahler metric on a projective manifold encounters a finite-time volume collapsing, this projective manifold admits a Fano fibration over a lower dimensional base. In this case, we prove the continuity method converges to a singular K\"ahler metric on the base in the weak sense; moreover, if the base is smooth and the fibration has no singular fibers, then the convergence takes place in Gromov-Hausdorff topology.