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ACC for foliated log canonical thresholds

Yen-An Chen

TL;DR

This paper extends the ascending chain condition (ACC) for log canonical thresholds to foliated settings in low dimensions. It defines foliated log canonical thresholds and develops a foliated minimal model program framework, including foliated dlt modifications and adjunction along invariant and non-invariant divisors, to control boundary coefficients. The authors prove a finiteness theorem: for dimensions $n\le 3$ and foliation rank $r<n$, the coefficients of boundary divisors in lc foliated triples with a common lc centre inside the boundary lie in a finite subset determined by the ambient coefficient set. Consequently, ACC for foliated LCT follows, covering rank-one foliations on surfaces and rank-one/rank-two foliations on threefolds, with detailed case analyses using adjunction, resolution of singularities, and MMP. The results align foliated birational geometry with established ACC results in the non-foliated setting for the crucial low-dimensional cases, enabling robust inductive approaches in the foliated MMP program.

Abstract

It is known that the set of log canonical thresholds (lcts) on any varieties with fixed dimension satisfies the ascending chain condition. Inspired by the foliated minimal model program, it is intriguing to study the foliated version of lcts and ask whether they have the similar property. We give an affirmative answer in the case of surfaces and threefolds.

ACC for foliated log canonical thresholds

TL;DR

This paper extends the ascending chain condition (ACC) for log canonical thresholds to foliated settings in low dimensions. It defines foliated log canonical thresholds and develops a foliated minimal model program framework, including foliated dlt modifications and adjunction along invariant and non-invariant divisors, to control boundary coefficients. The authors prove a finiteness theorem: for dimensions and foliation rank , the coefficients of boundary divisors in lc foliated triples with a common lc centre inside the boundary lie in a finite subset determined by the ambient coefficient set. Consequently, ACC for foliated LCT follows, covering rank-one foliations on surfaces and rank-one/rank-two foliations on threefolds, with detailed case analyses using adjunction, resolution of singularities, and MMP. The results align foliated birational geometry with established ACC results in the non-foliated setting for the crucial low-dimensional cases, enabling robust inductive approaches in the foliated MMP program.

Abstract

It is known that the set of log canonical thresholds (lcts) on any varieties with fixed dimension satisfies the ascending chain condition. Inspired by the foliated minimal model program, it is intriguing to study the foliated version of lcts and ask whether they have the similar property. We give an affirmative answer in the case of surfaces and threefolds.
Paper Structure (18 sections, 21 theorems, 22 equations)

This paper contains 18 sections, 21 theorems, 22 equations.

Table of Contents

  1. Preliminaries
  2. Foliations
  3. Indices on foliated surfaces
  4. Non-invariant curves
  5. Invariant curves
  6. Ascending/Descending chain condition
  7. Foliated dlt modification
  8. Foliations of rank one on surfaces
  9. Foliations of rank two on threefolds
  10. Adjunction
  11. ACC for LCT when n=3 and r=2
  12. E_1 is non-invariant
  13. E_1 is invariant
  14. Foliations of rank one on threefolds
  15. Resolution of singularities
  16. ...and 3 more sections

Key Result

Theorem 3

Fix $n\in{\mathbb N}$, $I\subset [0,1]$, and a subset $J$ of positive real numbers. If $I$ and $J$ both satisfy the descending chain condition, then the set satisfies the ascending chain condition where $M$ is ${\mathbb R}$-Cartier and $\mathfrak{T}_n(I)$ is the set of log canonical pairs $(X,\Delta)$ where $X$ is a normal variety of dimension $n$ and $\Delta\in I$.

Theorems & Definitions (56)

  • Conjecture 1: shokurov1988problems
  • Conjecture 2: ambro1999minimal
  • Theorem 3: hacon2014acc
  • Definition 4
  • Theorem 5
  • Theorem 6
  • Definition 1.1
  • Definition 1.2: Rational transform of foliations
  • Definition 1.3
  • Definition 1.4: Foliated pair
  • ...and 46 more