On the equivalence between the effective adjunction conjectures of Prokhorov–Shokurov and of Li

Han, Jingjun; Liu, Jihao; Xue, Qingyuan

doi:10.2140/ant.2025.19.2261

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On the equivalence between the effective adjunction conjectures of Prokhorov–Shokurov and of Li

Jingjun Han, Jihao Liu and Qingyuan Xue

Vol. 19 (2025), No. 11, 2261–2279

DOI: 10.2140/ant.2025.19.2261

Abstract

Prokhorov and Shokurov introduced the effective adjunction conjecture, also known as the effective basepoint-freeness conjecture, which asserts that the moduli component of an lc-trivial fibration is effectively basepoint-free. Li proposed a variation of this conjecture, known as the $Γ$ -effective adjunction conjecture, and demonstrated that a weaker version of his conjecture follows from the original Prokhorov–Shokurov conjecture.

In this paper, we prove the equivalence between Prokhorov–Shokurov’s and Li’s effective adjunction conjectures. The key to our proof is establishing uniform rational polytopes for canonical bundle formulas. This relies on recent advancements in the minimal model program theory of algebraically integrable foliations, primarily developed by Ambro–Cascini–Shokurov–Spicer and Chen–Han–Liu–Xie.

Keywords

algebraically integrable foliation, canonical bundle formula, uniform rational polytope

Mathematical Subject Classification

Primary: 14E30, 37F75

Milestones

Received: 23 December 2023

Revised: 31 July 2024

Accepted: 26 September 2024

Published: 14 September 2025

Authors

Jingjun Han
Shanghai Center for Mathematical Sciences Fudan University Shanghai China

Jihao Liu
Department of Mathematics Peking University Beijing China

Qingyuan Xue
Shanghai Center for Mathematical Sciences Fudan University Shanghai China

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Vol. 19, No. 11, 2025