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

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 $\Gamma$-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.