Finite generation of generalized klt log-canonical rings

Must the log-canonical ring of a projective klt generalized pair be finitely generated? Unlike an ordinary pair, a generalized pair includes a nef part; the recorded counterexample target uses a smooth complex projective 19-fold with B=0, K_X∼−16ξ and nef part M_X=17ξ for a nef Cartier divisor ξ.

Why it matters and what counts as progress

Why it matters: A non-finitely-generated example would show that adding a nef part can break finite-generation conclusions familiar from ordinary pairs.

What counts as progress: Construct a generalized klt pair with a non-finitely-generated ring or prove finite generation under stronger positivity conditions.

How the problem developed

2010 · Finite generation result [1]
Caucher Birkar, Paolo Cascini, Christopher Hacon and James McKernan Proved finite generation of canonical rings for smooth projective varieties of log general type, an important ordinary-pair result preceding the generalized-pair example.
4 Aug 2026 · Reported counterexample [2]
Jihao Liu and Yanze Wang Constructed a projective klt generalized pair whose generalized log-canonical ring is not finitely generated; the authors state that generative AI produced the main result.

AI contribution

A klt generalized pair with infinitely generated canonical ring

Jihao Liu, Yanze Wang

Liu and Wang construct a 19-dimensional smooth complex projective X with B=0, nef Cartier ξ, K_X linearly equivalent to −16ξ and nef part M_X descended to 17ξ. Its generalized log-canonical ring R(X,ξ), and consequently R(X,−K_X), is not finitely generated.

AI involvementAI-centered

ImportanceNotable

Technical verificationReported

Human understandingFew