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On multiplicities of fibers of Fano fibrations

Guodu Chen, Chuyu Zhou

Abstract

In this note, we reduce various conjectures in birational geometry, including Shokurov conjecture on singularities of the base of log Calabi-Yau fibrations of Fano type and boundedness conjecture for rationally connected Calabi-Yau varieties, to a conjecture on multiplicities of fibers of Fano fibrations over curves.

On multiplicities of fibers of Fano fibrations

Abstract

In this note, we reduce various conjectures in birational geometry, including Shokurov conjecture on singularities of the base of log Calabi-Yau fibrations of Fano type and boundedness conjecture for rationally connected Calabi-Yau varieties, to a conjecture on multiplicities of fibers of Fano fibrations over curves.
Paper Structure (15 sections, 29 theorems, 36 equations)

This paper contains 15 sections, 29 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Divisors
  4. Singularities of g-pairs
  5. Canonical bundle formulas
  6. Boundedness
  7. From multiplicities to singularities
  8. Proof of Theorem \ref{['thm1']}
  9. Generalized version of Conjecture \ref{['conj:SM']}
  10. On boundedness of RC Calabi-Yau varieties
  11. Conjectures on RC Calabi-Yau varieties
  12. Proof of Theorem \ref{['thm2']}
  13. Proof of Theorem \ref{['thm3']}
  14. Conjecture \ref{['conj:gcybdd']} in dimension $\le 3$
  15. Further remark on Conjecture \ref{['conj:rccyindex']}

Key Result

Theorem 1.6

(=Proposition 1.2 to 1.1) Conjecture conj:bddfiber in dimension $d$ implies Conjecture conj:SM in dimension $d$.

Theorems & Definitions (69)

  • Conjecture 1.1: Mckernan-Shokurov
  • Conjecture 1.2
  • Conjecture 1.3
  • Conjecture 1.4
  • Remark 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Definition 2.1
  • Definition 2.2
  • ...and 59 more