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On existence of bounded relative-global complements for Fano fibrations

Sung Rak Choi, Chuyu Zhou

Abstract

For Fano fibrations with $ε$-lc singularities of a fixed dimension, we show the existence of bounded relative-global complements. If the base of the fibration is of dimension one, we even show the existence of bounded relative-global klt complements.

On existence of bounded relative-global complements for Fano fibrations

Abstract

For Fano fibrations with -lc singularities of a fixed dimension, we show the existence of bounded relative-global complements. If the base of the fibration is of dimension one, we even show the existence of bounded relative-global klt complements.
Paper Structure (5 sections, 10 theorems, 77 equations)

This paper contains 5 sections, 10 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Global complement engine
  3. Curve base
  4. Higher dimensional base
  5. Final remark

Key Result

Theorem 1.2

(Birkar19) Let $d$ be a positive integer. Then there exists a positive integer $N$ depending only on $d$ satisfying the following: If $X\to Z$ is a Fano fibration of dimension $d$, then for any closed point $z\in Z$, there is an effective $\mathbb{Q}$-divisor $\Lambda$ on $X$ such that $N(K_X+\Lambd

Theorems & Definitions (18)

  • Conjecture 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 2.1
  • Definition 2.2
  • Theorem 2.3
  • Lemma 3.1
  • proof
  • ...and 8 more