Fano manifolds with weak almost K\"ahler-Ricci solitons

In this paper, we prove that a sequence of weak almost K"ahler-Ricci solitons under further suitable conditions converge to a K"ahler-Ricci soliton with complex codimension of singularities at least 2 in the Gromov-Hausdorff topology. As a corollary, we show that on a Fano manifold with the modified K-energy bounded below, there exists a sequence of weak almost K"ahler-Ricci solitons which converge to a K"ahler-Ricci soliton with complex codimension of singularities at least 2 in the Gromov-Hausdorff topology.