On diminished multiplier ideal and the termination of flips

TL;DR

This paper develops a theory of diminished multiplier ideals on singular varieties and proves the termination of certain flips with scaling under specific conditions, advancing the minimal model program.

Contribution

It introduces a new theory of diminished multiplier ideals and applies it to prove flip termination results with bounded Cartier index and specific Kodaira dimension conditions.

Findings

Proved flip termination under bounded Cartier index and $ abla_{ ext{sigma}}(K_X+ abla) extgreater{} ext{dim} X - 1$.

Developed a new framework of diminished multiplier ideals for singular varieties.

Connected multiplier ideal theory with flip termination in the minimal model program.

Abstract

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 $\kappa_{\sigma}(K_X+\Delta)\ge \dim X-1$ holds. The proof uses a theory of diminished multiplier ideal.