[Paper Review] Minimal model program for algebraically integrable foliations and generalized pairs

Guodu Chen, Jingjun Han+2 more|arXiv (Cornell University)|Sep 27, 2023

Nonlinear Waves and Solitons4 citations

TL;DR

This paper establishes the minimal model program (MMP) for $\mathbb{Q}$-factorial foliated dlt pairs and lc generalized pairs with algebraically integrable foliations, proving cone theorems, contraction theorems, and the existence of flips. It introduces generalized foliated quadruples and resolves a conjecture of Cascini and Spicer by showing that such foliations are induced by almost holomorphic maps under certain conditions.

ABSTRACT

By systematically introducing and studying the structure of algebraically integrable generalized foliated quadruples, we establish the minimal model program for $\mathbb Q$-factorial foliated dlt algebraically integrable foliations and lc generalized pairs by proving their cone theorems, contraction theorems, and the existence of flips. We also provide numerous applications on their birational geometry and resolve a conjecture of Cascini and Spicer.

Motivation & Objective

• To develop a minimal model program (MMP) for $\mathbb{Q}$-factorial foliated dlt pairs with algebraically integrable foliations.
• To extend the MMP to lc generalized pairs and generalized foliated quadruples, including canonical bundle formulas and ACC results.
• To resolve a conjecture by Cascini and Spicer on the structure of algebraically integrable foliations via MMP techniques.
• To establish the existence of good minimal models and Mori fiber spaces for such pairs under nefness and ampleness conditions.
• To prove the finite generation of canonical rings and base-point-freeness for relevant divisors in the generalized setting.

Proposed method

• Introduce and study the structure of algebraically integrable generalized foliated quadruples as a foundational framework.
• Prove the cone theorem and contraction theorem for $\mathbb{Q}$-factorial foliated dlt pairs via inductive arguments and bend-and-break techniques.
• Establish the existence of flips using MMP with scaling and super-divisors, leveraging boundedness and rational polytopes.
• Apply the canonical bundle formula for lc-trivial fibrations to derive adjunction and moduli part properties in the generalized setting.
• Use property $(*)$ and ACSS (almost canonical, semi-stable) models to control singularities and ensure termination of the MMP.
• Employ birational geometry tools such as dlt modifications, perturbations, and universal push-out diagrams for lc centers.

Experimental results

Research questions

• RQ1 Under what conditions does the minimal model program terminate for algebraically integrable foliations on $\mathbb{Q}$-factorial varieties?
• RQ2 Can the existence of flips and contractions be established for generalized pairs with algebraically integrable foliations?
• RQ3 Is the canonical ring of $K_{\mathcal{F}} + B + A$ finitely generated when $A$ is ample and $K_{\mathcal{F}} + B + A$ is nef?
• RQ4 Does the canonical bundle formula hold for generalized foliated quadruples, and what are its implications for moduli and discriminant parts?
• RQ5 When is a foliation induced by an almost holomorphic map, and how does this relate to the MMP and singularities?

Key findings

• The cone theorem, contraction theorem, and existence of flips hold for $\mathbb{Q}$-factorial projective F-dlt foliated triples with algebraically integrable foliations.
• If $K_{\mathcal{F}} + B + A$ is nef with $A$ ample, then $K_{\mathcal{F}} + B + A$ is semi-ample, confirming a special case of the base-point-freeness conjecture.
• When $B \geq A \geq 0$, the pair admits a good minimal model or a Mori fiber space, establishing the existence of good minimal models.
• The canonical ring $R(X, K_{\mathcal{F}} + B + A)$ is finitely generated when $K_{\mathcal{F}} + B + A$ is $\mathbb{Q}$-Cartier.
• For a generalized foliated quadruple with $a(D, \mathcal{F}, B, \mathbf{M}) > -1$ for all exceptional divisors, the foliation is induced by an almost holomorphic map.
• The global ACC and ACC for lc thresholds hold for generalized foliated quadruples, with uniform rational polytopes established via boundedness results.