Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Singularities on Fano fibrations and beyond

Caucher Birkar

TL;DR

This work advances the birational theory of fibrations by proving Shokurov’s conjecture on discriminant divisors for Fano type fibrations and establishing a stronger log Calabi–Yau fibration analogue, both via a synthesis of the MMP, complements, generalized pairs, and toroidal/toric methods. A key technical achievement is a multiplicity bound for special fibres along lc places, which the authors reduce to toric problems and solve, enabling global boundedness statements. As consequences, the paper derives boundedness up to codimension one for rationally connected varieties with nef $-K$, bounded klt complements for Fano fibrations over curves, and a Calabi–Yau index bound for RC Calabi–Yau pairs, with broader implications for moduli, arithmetic, and physics contexts. The techniques developed—toroidalisation, toric modelling, and strict-transform analysis—provide a robust toolkit for tackling singularities in families and have potential applications to F-theory, Kähler geometry, and beyond.

Abstract

In this paper, we investigate singularities on fibrations and related topics. We prove conjectures of McKernan and Shokurov on singularities on Fano type fibrations and a conjecture of the author on singularities on log Calabi-Yau fibrations. From these we derive a variant of a conjecture of McKernan and Prokhorov on rationally connected varieties with nef anti-canonical divisor. We present further applications to other problems including boundedness of klt complements for Fano fibrations over curves, torsion index of rationally connected Calabi-Yau pairs, and gonality of fibres of del Pezzo fibrations. We prove a general result on controlling multiplicities of fibres of certain fibrations (not necessarily of Fano type) which is the key ingredient of the proofs of the above results.

Singularities on Fano fibrations and beyond

TL;DR

This work advances the birational theory of fibrations by proving Shokurov’s conjecture on discriminant divisors for Fano type fibrations and establishing a stronger log Calabi–Yau fibration analogue, both via a synthesis of the MMP, complements, generalized pairs, and toroidal/toric methods. A key technical achievement is a multiplicity bound for special fibres along lc places, which the authors reduce to toric problems and solve, enabling global boundedness statements. As consequences, the paper derives boundedness up to codimension one for rationally connected varieties with nef , bounded klt complements for Fano fibrations over curves, and a Calabi–Yau index bound for RC Calabi–Yau pairs, with broader implications for moduli, arithmetic, and physics contexts. The techniques developed—toroidalisation, toric modelling, and strict-transform analysis—provide a robust toolkit for tackling singularities in families and have potential applications to F-theory, Kähler geometry, and beyond.

Abstract

In this paper, we investigate singularities on fibrations and related topics. We prove conjectures of McKernan and Shokurov on singularities on Fano type fibrations and a conjecture of the author on singularities on log Calabi-Yau fibrations. From these we derive a variant of a conjecture of McKernan and Prokhorov on rationally connected varieties with nef anti-canonical divisor. We present further applications to other problems including boundedness of klt complements for Fano fibrations over curves, torsion index of rationally connected Calabi-Yau pairs, and gonality of fibres of del Pezzo fibrations. We prove a general result on controlling multiplicities of fibres of certain fibrations (not necessarily of Fano type) which is the key ingredient of the proofs of the above results.
Paper Structure (43 sections, 28 theorems, 170 equations)

This paper contains 43 sections, 28 theorems, 170 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Morphisms
  4. Divisors, degree, and volume
  5. Pairs and singularities
  6. b-divisors
  7. Generalised pairs
  8. Minimal models, Mori fibre spaces, and MMP
  9. Fano type varieties
  10. Nakayama divisor
  11. Complements
  12. Adjunction for fibrations
  13. Image of divisors
  14. Coordinate hyperplanes
  15. Couples and toroidal geometry
  16. ...and 28 more sections

Key Result

Theorem 1.1

Let $d\in \mathbb{N}$ and let $\epsilon \in \mathbb{R}^{>0}$. Then there is $\delta \in \mathbb{R}^{>0}$ depending only on $d,\epsilon$ satisfying the following. Let $(X,B)$ be a pair where $B$ is a $\mathbb Q$-boundary and let $f\colon X\to Z$ be a contraction such that Then the generalised pair $(Z,B_Z+M_Z)$ induced by the canonical bundle formula is generalised $\delta$-lc.

Theorems & Definitions (54)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 44 more