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Weak transcendental base-point freeness and diameter lower bounds for the Kähler-Ricci flow

Junsheng Zhang

TL;DR

This work addresses diameter lower bounds along the Kähler–Ricci flow for non-Fano initial data by proving a weaker transcendental base-point freeness result. The authors develop a robust relative MMP framework for projective morphisms of compact Kähler spaces and establish pushforward/nef-pseudo-effective behavior of adjoint classes under extremal contractions and Mori–Fano fibrations, including generalized klt pairs. A central contribution is a weak base-point-freeness theorem that connects the adjoint class $K_X+\alpha$ to a fibration structure with a base whose dimension equals the numerical dimension $\mathrm{nd}(K_X+\alpha)$, with extensions to generalized pairs and nd$\le$3 cases. Finally, a generalized Schwarz lemma is proved and used in tandem with the base-point-freeness results to obtain a diameter lower bound for the Kähler–Ricci flow at the singular time, revealing a precise dichotomy between Fano and non-Fano behavior in finite-time extinction scenarios.

Abstract

We prove a weaker version of the transcendental base-point freeness on compact Kähler manifolds. As a consequence, we derive the diameter lower bound for finite time singularities of Kähler-Ricci flow with non-Fano initial data.

Weak transcendental base-point freeness and diameter lower bounds for the Kähler-Ricci flow

TL;DR

This work addresses diameter lower bounds along the Kähler–Ricci flow for non-Fano initial data by proving a weaker transcendental base-point freeness result. The authors develop a robust relative MMP framework for projective morphisms of compact Kähler spaces and establish pushforward/nef-pseudo-effective behavior of adjoint classes under extremal contractions and Mori–Fano fibrations, including generalized klt pairs. A central contribution is a weak base-point-freeness theorem that connects the adjoint class to a fibration structure with a base whose dimension equals the numerical dimension , with extensions to generalized pairs and nd3 cases. Finally, a generalized Schwarz lemma is proved and used in tandem with the base-point-freeness results to obtain a diameter lower bound for the Kähler–Ricci flow at the singular time, revealing a precise dichotomy between Fano and non-Fano behavior in finite-time extinction scenarios.

Abstract

We prove a weaker version of the transcendental base-point freeness on compact Kähler manifolds. As a consequence, we derive the diameter lower bound for finite time singularities of Kähler-Ricci flow with non-Fano initial data.

Paper Structure

This paper contains 11 sections, 17 theorems, 149 equations.

Table of Contents

  1. Introduction
  2. Relative MMP for projective morphisms
  3. Prelininaries
  4. Extremal contractions and $\mathbb Q$-factoriality
  5. Pushing-forward some classes in $H^{1,1}_{\mathop{\mathrm{BC}}\nolimits}$
  6. Bigness of boundary classes preserved after extremal contractions
  7. Relative MMP with scaling of a relatively big $\mathbb R$-line bundle
  8. A weak transcendental base-point freeness result
  9. Proof of Theorem \ref{['thm-weak basepointfree']}
  10. A general version and applications
  11. A generalized Schwarz lemma and proof of Theorem \ref{['thm--main']}

Key Result

Theorem 1.1

Suppose $(X, \omega_0)$ is not Fano and $\mathop{\mathrm{nef}}\nolimits_X(\omega_0)=1$. Let $\omega_t$ be the solution of the Kähler-Ricci-flow eq--ric flow. Then

Theorems & Definitions (39)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Lemma 2.8
  • ...and 29 more