On diminished multiplier ideal and the termination of flips

In this paper, we develop a theory of diminished multiplier ideals on singular varieties which was introduced by Hacon, and developed by Lehmann. We prove a result regarding the termination of certain type of flips with scaling of an ample divisor if the Cartier index is bounded, and if ${\kappa }_{\sigma }(K_X+\mathrm{\Delta })\ge \dim X-1$ holds. The proof uses a theory of diminished multiplier ideal.