Bounded klt complements for Fano contractions

What OpenAI claims

Proves the finite-rational-coefficient form of Shokurov’s bounded-klt-complement conjecture for ϵ-lc complex Fano-type pairs with nef anti-log-canonical divisor. For ϵ-lc Fano contractions over any algebraically closed characteristic-zero field, it gives klt complements near every base point, with index bounded only by dimension and positive rational ϵ. (Lean)

Papers (2)

1. Bounded klt complements for Fano contractions

   25 September 2026 · Preprint, not peer reviewed · PDF

   For fixed dimension d and positive rational ϵ, we prove that every ϵ-log-canonical Fano contraction over an algebraically closed field of characteristic zero admits, near each closed base point, a klt complement of index bounded only by d and ϵ. Over ℂ, we also obtain monotone klt complements for Fano type pairs with nef anti-log-canonical divisor and coefficients in a fixed finite rational set. This proves the finite-rational-coefficient form of Shokurov's bounded-klt-complement conjecture.

2. Uniform Cartier sections for Fano type contractions

   25 September 2026 · Preprint, not peer reviewed · PDF

   We prove the Cartier-divisor conjecture of Birkar and Shokurov for rational boundaries in characteristic zero and for real boundaries over ℂ. For an ϵ-lc Fano type contraction with positive-dimensional base and nef negative log canonical class, a Cartier divisor through any prescribed base point can be chosen with pullback log canonical threshold bounded below in terms of the dimension and ϵ alone.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.