Non-algebraicity of non-abundant foliations and abundance for adjoint foliated structures

TL;DR

The paper addresses when foliations on complex varieties are algebraic and how to classify them up to birational equivalence, with emphasis on adjoint foliated structures. Under the abundance conjecture in dimension $d$, it proves a non-algebraicity criterion: a log canonical foliation $ u(rak{F})\neq \kappa(rak{F})$ implies non-algebraic integrability for rank ≤ $d$, and, likewise, shows abundance and the existence of good minimal models or Mori fiber spaces for algebraically integrable adjoint foliated structures. In dimension $d=3$, these results hold unconditionally, and the paper also resolves the Lu–Wu problem on abundance for surface adjoint foliated structures not necessarily algebraically integrable. The approach combines adjoint foliations, generalized foliated quadruples, stable families, and $m{b}$-semi-ample techniques to develop a foliated minimal-model framework with controlled abundance phenomena.

Abstract

Assuming the abundance conjecture in dimension $d$, we establish a non-algebraicity criterion of foliations: any log canonical foliation of rank $\le d$ with $ν\neqκ$ is not algebraically integrable, answering question of Ambro--Cascini--Shokurov--Spicer. Under the same hypothesis, we prove abundance for klt algebraically integrable adjoint foliated structures of dimension $\le d$ and show the existence of good minimal models or Mori fiber spaces. In particular, when $d=3$, all these results hold unconditionally. Using similar arguments, we solve a problem proposed by Lu and Wu on abundance of surface adjoint foliated structures that are not necessarily algebraically integrable.

Theorems & Definitions (71)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Theorem 1.4
• Definition 2.1
• Definition 2.2
• Definition 2.4
• Definition 2.5
• Definition 2.6: Foliations, cf. ACSS21CS21
• Definition 2.7