Log canonicity of the Proj of a finitely generated log canonical ring

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Let (X,Δ)(X,\Delta) be a projective log canonical pair. Assume that the log canonical ring

R(X,Δ)=⨁m≥0H0(X,OX(⌊m(KX+Δ)⌋))R(X,\Delta)=\bigoplus_{m\geq 0}H^0(X,\mathcal O_X(\lfloor m(K_X+\Delta)\rfloor))

is a finitely generated C\mathbb C-algebra, and set

Y=Proj⁡R(X,Δ).Y=\operatorname{Proj} R(X,\Delta).

The log canonical Proj conjecture. There is an effective Q\mathbb Q-divisor ΔY\Delta_Y on YY such that (Y,ΔY)(Y,\Delta_Y) is log canonical. The result is known when KYK_Y is Q\mathbb Q-Cartier, and the klt case is also established; the general log canonical statement remains open in the source.

References

Primary source

Osamu Fujino, “Some remarks on the minimal model program for log canonical pairs”, arXiv:1309.3015 (2014).

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