Complex Algebraic Geometry

Main conjectures of (log) MMP are: A) existence of flips, B) termination of flips, and C) finite generation, all of which were settled by 1992 in dim 3 by Mori, Shokurov, Kawamata, Koll\'ar et al. In dim 4, A was done by 2005 due to Shokurov and Hacon-McKernan. I will explain the following new results: B in dim 4 under a mild assumption, A and C in dim 5 in general. This is a joint work with Hacon and Kawamata.