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On diminished multiplier ideal and the termination of flips

Donghyeon Kim

TL;DR

The paper addresses termination of flips in the Minimal Model Program by developing diminished multiplier ideals on singular varieties and leveraging a Nadel-type vanishing framework. The authors extend the diminished multiplier ideal theory to klt pairs, establish a valuation-theoretic bridge to log canonical thresholds, and prove a termination result for flips with scaling under the condition $\kappa_{\sigma}(K_X+\Delta)\ge \dim X-1$ with bounded Cartier index. Key contributions include a generalization of diminished multiplier ideals to singular settings, a vanishing theorem, and a termination criterion that connects cohomological invariants to the dynamics of flips. This provides new tools for tackling long-standing questions in birational geometry and has potential to illuminate broader termination phenomena within the MMP.

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 $κ_σ(K_X+Δ)\ge \dim X-1$ holds. The proof uses a theory of diminished multiplier ideal.

On diminished multiplier ideal and the termination of flips

TL;DR

The paper addresses termination of flips in the Minimal Model Program by developing diminished multiplier ideals on singular varieties and leveraging a Nadel-type vanishing framework. The authors extend the diminished multiplier ideal theory to klt pairs, establish a valuation-theoretic bridge to log canonical thresholds, and prove a termination result for flips with scaling under the condition with bounded Cartier index. Key contributions include a generalization of diminished multiplier ideals to singular settings, a vanishing theorem, and a termination criterion that connects cohomological invariants to the dynamics of flips. This provides new tools for tackling long-standing questions in birational geometry and has potential to illuminate broader termination phenomena within the MMP.

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 holds. The proof uses a theory of diminished multiplier ideal.
Paper Structure (8 sections, 27 theorems, 153 equations)

This paper contains 8 sections, 27 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Valuation
  4. Log canonical threshold and asymptotic multiplier ideals
  5. Valuation and log canonical threshold
  6. Jonsson-Mustață theorem
  7. Diminished multiplier ideal and vanishing theorem
  8. Proof of the main theorem on termination of flips

Key Result

Theorem 1.1

Let $(X,\Delta)$ be a projective klt pair, and $D$ a big $\mathbb{Q}$-divisor. Then we have

Theorems & Definitions (65)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1: cf. Nak04
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Definition 2.5
  • ...and 55 more