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

Abstract

Prokhorov and Shokurov introduced the famous effective adjunction conjecture, also known as the effective base-point-freeness conjecture. This conjecture asserts that the moduli component of an lc-trivial fibration is effectively base-point-free. Li proposed a variation of this conjecture, which is known as the $\Gamma$-effective adjunction conjecture, and proved that a weaker version of his conjecture is implied by the original Prokhorov-Shokurov conjecture. In this paper, we establish the equivalence of Prokhorov-Shokurov's and Li's effective adjunction conjectures. The key to our proof is the formulation of a uniform rational polytope for canonical bundle formulas, which relies on recent developments in the minimal model program theory of algebraically integrable foliations by Ambro-Cascini-Shokurov-Spicer and Chen-Han-Liu-Xie.