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Boundedness of complements for fibered Fano threefolds in positive characteristic

Xintong Jiang

TL;DR

This work addresses boundedness of complements for Fano-type threefolds in large positive characteristic by establishing a canonical bundle formula for Fano-type fibrations and proving Shokurov's boundedness of complements in this setting. The authors adapt and extend the characteristic-zero strategy to positive characteristic, leveraging a restricted vanishing theorem, adjunctions, and the canonical bundle formula to control singularities and positivity across the base. A key innovation is the effective canonical bundle formula for Fano-type fibrations and the use of Frobenius-stable sections to lift complements, which leads to both relative and global boundedness results for complements in dimension three. The results contribute to the understanding and potential resolution of BAB-type boundedness in positive characteristic and provide tools for constructing bounded moduli of Fano-type fibrations in large characteristic regimes.

Abstract

In this paper, we prove the canonical bundle formula for Fano type fibrations and Shokurov's conjecture on boundedness of complements for Fano type threefold pairs $(X,B)$ with fibration structures in large characteristics. In particular, we prove the conjecture when $-(K_X+B)\not\equiv 0$ is nef and not big in large characteristics.

Boundedness of complements for fibered Fano threefolds in positive characteristic

TL;DR

This work addresses boundedness of complements for Fano-type threefolds in large positive characteristic by establishing a canonical bundle formula for Fano-type fibrations and proving Shokurov's boundedness of complements in this setting. The authors adapt and extend the characteristic-zero strategy to positive characteristic, leveraging a restricted vanishing theorem, adjunctions, and the canonical bundle formula to control singularities and positivity across the base. A key innovation is the effective canonical bundle formula for Fano-type fibrations and the use of Frobenius-stable sections to lift complements, which leads to both relative and global boundedness results for complements in dimension three. The results contribute to the understanding and potential resolution of BAB-type boundedness in positive characteristic and provide tools for constructing bounded moduli of Fano-type fibrations in large characteristic regimes.

Abstract

In this paper, we prove the canonical bundle formula for Fano type fibrations and Shokurov's conjecture on boundedness of complements for Fano type threefold pairs with fibration structures in large characteristics. In particular, we prove the conjecture when is nef and not big in large characteristics.

Paper Structure

This paper contains 23 sections, 38 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Preliminaries
  4. Resolution of singularities
  5. Hyperstandard sets
  6. Divisors, pairs and linear systems
  7. B-divisors and generalized pairs
  8. Singularities from MMP
  9. Contractions and minimal model programs
  10. $\mathbb Q$-factorialization
  11. Volumes, Kodaira dimensions and Iitaka fibrations
  12. Fano pairs and varieties of Fano type
  13. Bounded families
  14. Complements
  15. Vanishing theorem for pl-contraction of plt pairs
  16. ...and 8 more sections

Key Result

Theorem 1.3

Let $R\subset [0,1]$ be a finite set of rational numbers, there is an $n\in\mathbb N$ and a prime number $p_0$ depending only on $R$ such that if $(X,B)$ is a projective lc pair defined over an algebraically closed field $k$ satisfying the following conditions: then there is an $n$-complement $(X,B^+)$ for $(X,B)$ with $B^+\geq B$.

Theorems & Definitions (72)

  • Conjecture 1.1: BAB Conjecture
  • Conjecture 1.2: Boundedness of complements, Shokurov
  • Theorem 1.3: \ref{['fibcomp']}
  • Theorem 1.4: \ref{['fibcomp2']}
  • Theorem 1.5: Canonical bundle formula for Fano type fibrations, \ref{['cbf1']}
  • Corollary 1.6: Contraction of Fano type threefolds is of Fano type, \ref{['contr']}
  • Theorem 1.7: \ref{['recomp']}
  • Theorem 2.1
  • proof
  • Theorem 2.2: cf. KYH20lcmmp
  • ...and 62 more