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On Effective Iitaka Fibration Indices for Stable Minimal Models with Large Iitaka Volumes

Hexu Liu

Abstract

Given a pair $(V,C)$ that admits a stable minimal model with fixed dimension, fixed coefficient set, and bounded relative volume, we study when the linear system $|m(K_V+C)|$ induces an Iitaka fibration, assuming the Iitaka volume of $K_V+C$ is sufficiently large.

On Effective Iitaka Fibration Indices for Stable Minimal Models with Large Iitaka Volumes

Abstract

Given a pair that admits a stable minimal model with fixed dimension, fixed coefficient set, and bounded relative volume, we study when the linear system induces an Iitaka fibration, assuming the Iitaka volume of is sufficiently large.

Paper Structure

This paper contains 23 sections, 34 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. DCC sets.
  4. Divisors and contractions.
  5. Linear systems.
  6. Pairs and Singularities.
  7. B-divisors and Generalized Pairs
  8. Iitaka dimension and Iitaka volumes.
  9. Stable minimal models.
  10. Nakayama-Zariski decomposition of divisors.
  11. Adjunction formula for fibrations.
  12. Excellent models, strong excellent models for adjunction.
  13. Boundedness properties for stable minimal models.
  14. Adjunction for lc centers.
  15. Kollár's Injectivity Lemma.
  16. ...and 8 more sections

Key Result

Theorem 1.2

Let $n>0$ be an integer, $\Phi \subseteq [0,1]$ be a DCC set, $u>0$ be a real number. Then there exists an integer $m_0 > 0$ such that the following holds. Assume that $(V,C)$ admits a klt $(n,\Phi,{\leq}u)$-stable minimal model (see Section stable-minimal-models for definitions). Then, for any posi

Theorems & Definitions (88)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Definition 2.1
  • Definition 2.2
  • ...and 78 more