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A note on lc-trivial fibrations

Kenta Hashizume

Abstract

For every lc-trivial fibration $(X,Δ) \to Z$ from an lc pair, we prove that after a base change, there exists a positive integer $n$, depending only on the dimension of $X$, the Cartier index of $K_{X}+Δ$, and the sufficiently general fibers of $X \to Z$, such that $n(K_{X}+Δ)$ is linearly equivalent to the pullback of a Cartier divisor.

A note on lc-trivial fibrations

Abstract

For every lc-trivial fibration from an lc pair, we prove that after a base change, there exists a positive integer , depending only on the dimension of , the Cartier index of , and the sufficiently general fibers of , such that is linearly equivalent to the pullback of a Cartier divisor.
Paper Structure (4 sections, 9 theorems, 30 equations)

This paper contains 4 sections, 9 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Definitions
  3. Proofs of main results
  4. Lc-trivial fibration with log big moduli part

Key Result

Theorem 1.1

For every $d$, $m \in \mathbb{Z}_{>0}$ and $v \in \mathbb{R}_{>0}$, there exists $n \in \mathbb{Z}_{>0}$, depending only on $d$, $m$, and $v$, satisfying the following. Let $(X,\Delta)$ be a projective lc pair, let $\pi \colon (X,\Delta) \to Z$ be an lc-trivial fibration with the sufficiently genera Then there exists a generalized lc pair $(Z,\boldsymbol{\rm B}_{Z}+\boldsymbol{\rm M})$ defined wit

Theorems & Definitions (23)

  • Theorem 1.1: Main result
  • Theorem 1.2
  • Theorem 1.3: floris-lazic
  • Theorem 1.4: Theorem \ref{['thm--lctrivialfibration-boundedfiber-abundance']} and Theorem \ref{['thm--lctrivialfibration-boundedfiber-antiabundance']}
  • Theorem 1.5: =Theorem \ref{['thm--minimalmodeltheory-logbigmoduli']}
  • Definition 2.1: b-divisor
  • Definition 2.2: Generalized lc pair
  • Definition 2.3: Lc-trivial fibration for lc pair
  • Definition 2.4: Discriminant b-divisor, moduli b-divisor, canonical bundle formula
  • Definition 2.5: Ambro model and log smooth Ambro model
  • ...and 13 more