Li's Γ\Gamma-effective adjunction conjecture

Let dd be a positive integer and let Γ[0,1]\Gamma\subset [0,1] be a DCC set of real numbers. A Γ\Gamma-base-point-free bb-divisor D\mathbf{D} on a normal projective variety XX is one for which there exist a1,,akΓa_1,\ldots,a_k\in\Gamma and base-point-free bb-divisors D1,,Dk\mathbf{D}_1,\ldots,\mathbf{D}_k such that

i=1kai=1,D=i=1kaiDi.\sum_{i=1}^k a_i=1,\qquad \mathbf{D}=\sum_{i=1}^k a_i\mathbf{D}_i.

Let f:(X,B)Zf:(X,B)\rightarrow Z be an lc-trivial fibration with dimXdimZ=d\dim X-\dim Z=d, whose horizontal/Z/Z coefficients belong to Γ\Gamma, and let M\mathbf{M} be its moduli part. Li's conjecture. There exist a positive integer II and a finite set Γ0(0,1]\Gamma_0\subset(0,1], depending only on dd and Γ\Gamma, such that IMI\mathbf{M} is Γ0\Gamma_0-base-point-free. This is a stronger irrational-coefficient variation of the Prokhorov–Shokurov conjecture; the paper establishes the equivalence of the two effective adjunction conjectures, but the supplied text does not state that this conjecture itself is resolved.

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Primary source

Jingjun Han, Jihao Liu and Qingyuan Xue, “On the equivalence between the effective adjunction conjectures of Prokhorov-Shokurov and of Li”, arXiv:2312.15397 (2023).

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