Finite generation conjecture for log canonical rings of simple normal crossing pairs

Let XX be a smooth projective variety and let Δ\Delta be an effective Q\mathbb Q-divisor on XX such that SuppΔ\operatorname{Supp}\Delta is a simple normal crossing divisor and the coefficients of Δ\Delta are less than or equal to one.

Finite generation conjecture for log canonical rings. The log canonical ring

R(X,KX+Δ)=m=0H0(X,OX(m(KX+Δ)))R(X,K_X+\Delta)=\bigoplus_{m=0}^{\infty}H^0\left(X,\mathcal O_X\left(\lfloor m(K_X+\Delta)\rfloor\right)\right)

is a finitely generated C\mathbb C-algebra.

This generalizes the finite-generation theorem for the corresponding klt setting by allowing coefficients equal to one. The source discusses its relationship with other minimal-model-program conjectures but does not state a resolution here.

Sources & referencesView supporting material

Primary source

Osamu Fujino, “On semipositivity, injectivity and vanishing theorems”, arXiv:1503.06503 (2016).

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