On global ACC for foliated threefolds

Abstract

In this paper, we prove the rational coefficient case of the global ACC for foliated threefolds. Specifically, we consider any lc foliated log Calabi-Yau triple $(X,\mathcal{F},B)$ of dimension $3$ whose coefficients belong to a set $\Gamma$ of rational numbers satisfying the descending chain condition, and prove that the coefficients of $B$ belong to a finite set depending only on $\Gamma$. To prove our main result, we introduce the concept of generalized foliated quadruples, which is a mixture of foliated triples and Birkar-Zhang's generalized pairs. With this concept, we establish a canonical bundle formula for foliations in any dimension. As for applications, we extend Shokurov's global index conjecture in the classical MMP to foliated triples and prove this conjecture for threefolds with nonzero boundaries and for surfaces. Additionally, we introduce the theory of rational polytopes for functional divisors on foliations and prove some miscellaneous results.Comment: 22 pages. Add a paragraph on pages 3-4. Proposition 6.4 and Lemma 7.2 strengthened. Small modification of the proof of 8.1. Reference update