Finite-coefficient conjecture for numerically trivial generalized pairs

Let dd be a positive integer and let Γ[0,+)\Gamma\subset [0,+\infty) be a DCC set. A projective lc generalized pair (X,B,M)(X,B,\mathbf{M}) consists here of a projective generalized pair of dimension dd that is log canonical. Assume that BΓB\in\Gamma, MNef0(Γ)\mathbf{M}\in\mathrm{Nef}^0(\Gamma), and

KX+B+MX0.K_X+B+\mathbf{M}_X\equiv 0.

Finite-coefficient conjecture. There exists a finite set Γ0Γ\Gamma_0\subset\Gamma, depending only on dd and Γ\Gamma, such that BΓ0B\in\Gamma_0 and MNef0(Γ0)\mathbf{M}\in\mathrm{Nef}^0(\Gamma_0).

This is a boundedness statement for the coefficients and nef-part coefficients of numerically trivial projective lc generalized pairs in fixed dimension. The supplied text gives no resolution status or further evidence, so the conjecture is recorded as open.

Sources & referencesView supporting material

Primary source

Tianle Yang, Zelin Ye and Zhiyao Zhang, “Existence of minimal models for threefold generalized pairs in positive characteristic”, arXiv:2408.12269 (2026).

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