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Variation of algebraically integrable adjoint foliated structures

Paolo Cascini, Jihao Liu, Fanjun Meng, Roberto Svaldi, Lingyao Xie

Abstract

Given a canonical algebraically integrable foliation on a klt projective variety, we study the variation of the ample models of the associated adjoint foliated structures with respect to the parameter. When the foliation is of general type, we show the finiteness of ample models if the parameter is sufficiently close to $1$. When the ambient variety is of general type, we show the finiteness of ample models for all parameters. A key ingredient in our proof is the equivalence between the existence of minimal models and the termination of MMP with scaling for algebraically integrable adjoint foliated structures.

Variation of algebraically integrable adjoint foliated structures

Abstract

Given a canonical algebraically integrable foliation on a klt projective variety, we study the variation of the ample models of the associated adjoint foliated structures with respect to the parameter. When the foliation is of general type, we show the finiteness of ample models if the parameter is sufficiently close to . When the ambient variety is of general type, we show the finiteness of ample models for all parameters. A key ingredient in our proof is the equivalence between the existence of minimal models and the termination of MMP with scaling for algebraically integrable adjoint foliated structures.

Paper Structure

This paper contains 9 sections, 24 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Adjoint foliated structures
  5. Models of adjoint foliated structures
  6. Existence of minimal models and MMP with scaling
  7. Finiteness of ample models
  8. Basic properties of ample models
  9. Proof of the main theorems

Key Result

Theorem 1.1

Let $X$ be a klt projective variety and $\mathcal{F}$ a canonical algebraically integrable foliation on $X$. For $t\in[0,1]$ set ${\mathfrak{A}}_t:=(X,\mathcal{F},t)$. Assume that ${\mathfrak{A}}_{\lambda}$ is of general type for some $\lambda\in [0,1]$. Then there exist a rational number $\epsilon< with the following properties. Let $\Gamma:=\{\{t_i\},\,(t_i,t_{i+1})\mid 1\leq i\leq n-1\}$. For a

Theorems & Definitions (70)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.3
  • Definition 2.4: NQC
  • Definition 2.5
  • Definition 2.6: Foliations, cf. ACSS21CS21
  • Definition 2.7
  • Definition 2.8: Qdlt, Cas+24
  • Definition 2.10: Potentially klt
  • ...and 60 more