Uniform index for numerically trivial foliated triples

For projective log-canonical foliated triples of dimension at most three with numerically trivial K_F+B, can one bound the index making it linearly trivial? Boundary coefficients belong to a fixed rational descending-chain-condition set.

Why it matters and what counts as progress

Why it matters: A uniform index turns numerical triviality into bounded torsion for foliated adjoint divisors in dimension at most three.

What counts as progress: Find one uniform I with I(K_F+B) linearly trivial, including non-algebraically-integrable foliations, or determine the dependence on the coefficient set.

How the problem developed

23 Mar 2023 · Partial result [1]
Jihao Liu, Fanjun Meng and Lingyao Xie Proved foliated global-index statements for threefolds with nonzero boundary and for surfaces, leaving the canonical threefold case as a central gap.
21 May 2026 · Reported claim [2]
Jihao Liu and Sheng Qin Reported a uniform index bound for log-canonical foliated triples in dimension at most three, including the numerically trivial canonical threefold case.

AI contribution

Shokurov's global index conjecture for threefold foliations

Jihao Liu, Sheng Qin

Liu and Qin prove a uniform index I for projective log-canonical foliated triples of dimension at most three with K_F+B numerically trivial and coefficients in a fixed rational DCC set, so I(K_F+B) is linearly trivial. Rethlas contributed the non-algebraically-integrable case split and Lie-theoretic route.

AI involvementAI-centered

ImportanceNotable

Technical verificationFormal (human)

Human understandingFull