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Fundamental properties of basic slc-trivial fibrations, II

Osamu Fujino, Taro Fujisawa, Haidong Liu

TL;DR

The moduli part of a basic slc-trivial fibration is semi-ample when the base space is a curve because the moduli of the divisor is b-numerically trivial.

Abstract

We prove that if the moduli $\mathbb Q$-b-divisor of a basic slc-trivial fibration is b-numerically trivial then it is $\mathbb Q$-b-linearly trivial. As a consequence, we prove that the moduli part of a basic slc-trivial fibration is semi-ample when the base space is a curve.

Fundamental properties of basic slc-trivial fibrations, II

TL;DR

The moduli part of a basic slc-trivial fibration is semi-ample when the base space is a curve because the moduli of the divisor is b-numerically trivial.

Abstract

We prove that if the moduli -b-divisor of a basic slc-trivial fibration is b-numerically trivial then it is -b-linearly trivial. As a consequence, we prove that the moduli part of a basic slc-trivial fibration is semi-ample when the base space is a curve.

Paper Structure

This paper contains 5 sections, 8 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quick review of basic slc-trivial fibrations
  4. On variation of mixed Hodge structure
  5. Proof of Theorem \ref{['B-thm1.3']}

Key Result

Theorem 1.1

Let $f\colon (X, B)\to Y$ be a basic slc-trivial fibration and let $\mathbf B$ and $\mathbf M$ be the induced discriminant and moduli $\mathbb Q$-b-divisors of $Y$ respectively. Then we have the following properties:

Theorems & Definitions (25)

  • Theorem 1.1: fujino-slc-trivial
  • Conjecture 1.2: b-semi-ampleness conjecture, see fujino-slc-trivial
  • Theorem 1.3: Main Theorem
  • Corollary 1.4
  • Definition 2.1: Simple normal crossing pairs
  • Definition 2.2: Semi-log canonical pairs
  • Definition 2.4: Potentially nef divisors, see fujino-slc-trivial
  • Definition 2.5: see fujino-slc-trivial
  • Definition 3.1: Basic slc-trivial fibrations, see fujino-slc-trivial
  • Theorem 4.1: fujino-fujisawa
  • ...and 15 more