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On semi-ampleness of the moduli part

Stefano Filipazzi, Calum Spicer

TL;DR

The paper advances Shokurov's conjecture on the semi-ampleness of the moduli part $M_X$ of a GLC fibration by combining foliation theory with stability concepts. It proves a 1-dimensional-base case via a Simpson-theoretic approach and provides a general framework in which the semi-ampleness of $ ext{det } f_*igl( ext{O}_X(mM)igr)$ and the semi-ampleness of $M+oldsymbol{ extepsilon} f^*c_1( ext{det } f_*igl( ext{O}_X(mM)igr))$ follow from the b-semi-ampleness conjecture in relative dimension up to $n-1$, together with base-change compatibility. It also ties the Kodaira dimension of the moduli part to the variation of the fibration and shows that, under suitable hypotheses, the moduli part nearly determines the fibration's geometry, including isotrivial or product structures after finite base changes. Overall, the work delineates conditions under which the moduli part attains semi-ampleness and clarifies the role of stability, foliations, and canonical bundle formulas in controlling variation and birational models.

Abstract

We discuss a conjecture of Shokurov on the semi-ampleness of the moduli part of a general fibration.

On semi-ampleness of the moduli part

TL;DR

The paper advances Shokurov's conjecture on the semi-ampleness of the moduli part of a GLC fibration by combining foliation theory with stability concepts. It proves a 1-dimensional-base case via a Simpson-theoretic approach and provides a general framework in which the semi-ampleness of and the semi-ampleness of follow from the b-semi-ampleness conjecture in relative dimension up to , together with base-change compatibility. It also ties the Kodaira dimension of the moduli part to the variation of the fibration and shows that, under suitable hypotheses, the moduli part nearly determines the fibration's geometry, including isotrivial or product structures after finite base changes. Overall, the work delineates conditions under which the moduli part attains semi-ampleness and clarifies the role of stability, foliations, and canonical bundle formulas in controlling variation and birational models.

Abstract

We discuss a conjecture of Shokurov on the semi-ampleness of the moduli part of a general fibration.
Paper Structure (12 sections, 32 theorems, 17 equations)

This paper contains 12 sections, 32 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. (Locally) stable families
  4. Property $(*)$, BP stable and all that
  5. Preliminaries on foliations
  6. Numerical flatness to isotriviality
  7. Harder--Narasimhan filtrations
  8. Property $(*)$ pairs and locally stable families
  9. On semi-ampleness of the moduli part
  10. Stable pairs
  11. Intermediate Kodaira dimension
  12. $1$-dimensional bases

Key Result

Theorem 1.2

Let $f\colon (X, B) \rightarrow Z$ be a GLC fibration between projective varieties with $B \geq 0$, $\operatorname{dim} X= n$ and $\operatorname{dim} Z = 1$. Suppose that $(X, B)$ is klt over the generic point of $Z$ and the generic fibre of $f$ admits a good minimal model. Then we may find a birati

Theorems & Definitions (69)

  • Conjecture 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5: = Theorem \ref{['thm_*']}
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Lemma 2.4
  • ...and 59 more