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On finite generation and boundedness of adjoint foliated structures

Paolo Cascini, Jingjun Han, Jihao Liu, Fanjun Meng, Calum Spicer, Roberto Svaldi, Lingyao Xie

TL;DR

The paper develops a comprehensive MMP framework for adjoint foliated structures $(X, rak{F},B,f{M},t)$, introducing the canonical divisor $K_{ rak{A}}=tK_{ rak{F}}+(1-t)K_X+B+f{M}_X$ and proving two core results: (i) good minimal models exist for klt algebraically integrable adjoint foliated structures of general type (and finite generation of the canonical ring when $t$ is rational); and (ii) a boundedness result for Fano adjoint foliated structures with bounded total minimal log discrepancies and $t$ away from $0$ and $1$. The authors establish an MMP with scaling in this setting, prove base-point-freeness and contraction theorems, and develop finiteness and polyhedrality results (Shokurov-type polytopes) that enable finite models and connections via flops. A key application shows that ambient varieties of lc Fano algebraically integrable foliations are of Fano type when the ambient is potentially klt, and they formulate a foliated Borisov–Alexeev–Borisov-type boundedness (foliated BAB). The results collectively extend the BCHM framework to foliations, enabling finite generation, boundedness, and moduli-type insights for adjoint foliated structures in higher dimensions.

Abstract

We prove the existence of good minimal models for any klt algebraically integrable adjoint foliated structure of general type, and that Fano algebraically integrable adjoint foliated structures with total minimal log discrepancies and parameters bounded away from zero form a bounded family. These results serve as the algebraically integrable foliation analogues of the finite generation of the canonical rings proved by Birkar-Cascini-Hacon-M\textsuperscript{c}Kernan, and the Borisov-Alexeev-Borisov conjecture on the boundedness of Fano varieties proved by Birkar, respectively. As an application, we prove that the ambient variety of any lc Fano algebraically integrable foliation is of Fano type, provided the ambient variety is potentially klt.

On finite generation and boundedness of adjoint foliated structures

TL;DR

The paper develops a comprehensive MMP framework for adjoint foliated structures , introducing the canonical divisor and proving two core results: (i) good minimal models exist for klt algebraically integrable adjoint foliated structures of general type (and finite generation of the canonical ring when is rational); and (ii) a boundedness result for Fano adjoint foliated structures with bounded total minimal log discrepancies and away from and . The authors establish an MMP with scaling in this setting, prove base-point-freeness and contraction theorems, and develop finiteness and polyhedrality results (Shokurov-type polytopes) that enable finite models and connections via flops. A key application shows that ambient varieties of lc Fano algebraically integrable foliations are of Fano type when the ambient is potentially klt, and they formulate a foliated Borisov–Alexeev–Borisov-type boundedness (foliated BAB). The results collectively extend the BCHM framework to foliations, enabling finite generation, boundedness, and moduli-type insights for adjoint foliated structures in higher dimensions.

Abstract

We prove the existence of good minimal models for any klt algebraically integrable adjoint foliated structure of general type, and that Fano algebraically integrable adjoint foliated structures with total minimal log discrepancies and parameters bounded away from zero form a bounded family. These results serve as the algebraically integrable foliation analogues of the finite generation of the canonical rings proved by Birkar-Cascini-Hacon-M\textsuperscript{c}Kernan, and the Borisov-Alexeev-Borisov conjecture on the boundedness of Fano varieties proved by Birkar, respectively. As an application, we prove that the ambient variety of any lc Fano algebraically integrable foliation is of Fano type, provided the ambient variety is potentially klt.

Paper Structure

This paper contains 21 sections, 59 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Further discussion of the main results and sketch of the proofs
  3. MMP and good minimal models for adjoint foliated structures
  4. Birational geometry for algebraically integrable adjoint foliated structures
  5. MMP for algebraically integrable foliations on potentially klt varieties
  6. Foliated BAB
  7. Geography of minimal models
  8. Sketch of the proof of Theorem \ref{['thm: eogmm ai afs general case']}
  9. Preliminaries
  10. Notation
  11. Adjoint foliated structures
  12. Birational maps in the MMP
  13. Relative Nakayama-Zariski decompositions
  14. Perturbation results
  15. Families of foliations
  16. ...and 6 more sections

Key Result

Theorem A

Let $X$ be a smooth projective variety and $\mathcal{F}$ an algebraically integrable foliation on $X$ with at worst log canonical singularities. Let $t\in [0,1)$ be a real number such that $K:=tK_{\mathcal{F}}+(1-t)K_X$ is big. Then $K$ has a good minimal model. In particular, if $t$ is a rational n is finitely generated.

Theorems & Definitions (131)

  • Theorem A
  • Theorem B
  • Theorem C
  • Conjecture 2.1: Existence of good minimal models
  • Theorem 2.1.1
  • Theorem 2.1.2: Base-point-freeness theorem
  • Theorem 2.1.3: Contraction theorem
  • Theorem 2.1.4: Existence of MMP
  • Theorem 2.1.5: Existence of Mori fiber spaces
  • Theorem 2.1.6
  • ...and 121 more