Bounded klt complements for Fano contractions

2 · Going further

Fix a dimension $d$ and a positive rational number $ϵ$. The main manuscript shows that every $ϵ$-log-canonical Fano contraction over an algebraically closed field of characteristic zero admits a klt complement near each closed base point, with index bounded only by $d$ and $ϵ$. The statement is local near the chosen base point; the abstract does not assert that one complement works globally over the entire base.

Over $\mathbb{C}$, the main manuscript also obtains monotone klt complements for $ϵ$-lc Fano type pairs whose anti-log-canonical divisor is nef and whose boundary coefficients belong to a fixed finite set of rational numbers. It identifies this as the finite-rational-coefficient form of Shokurov’s bounded-klt-complement conjecture.

The Lean-formalized companion, “Uniform Cartier sections for Fano type contractions,” addresses the Cartier-divisor conjecture of Birkar and Shokurov for rational boundaries in characteristic zero and for real boundaries over $\mathbb{C}$. For an $ϵ$-lc Fano type contraction with positive-dimensional base and nef negative log canonical class, it gives a Cartier divisor through any prescribed base point whose pullback has log canonical threshold bounded below in terms of dimension and $ϵ$ alone.

The abstracts thus present two related forms of uniform control: bounded complement indices and a lower bound on the singularity threshold of chosen divisors. They do not describe the argument or specify how the divisor statement is used to obtain complements, so no more detailed proof mechanism can be extracted from the supplied text.

Useful notions

• Algebraic varieties and singularities
• Pairs and boundary coefficients
• Contractions and their bases
• Divisors and Cartier divisors
• Log canonical singularities and thresholds

What was known before / what is new

Limits and assumptions

The contraction statement assumes an algebraically closed field of characteristic zero, fixed dimension, and positive rational $ϵ$, and gives complements near each closed base point. The complex pair statement requires Fano type, the stated singularity bound, a nef anti-log-canonical divisor, and coefficients in a fixed finite rational set. The companion’s threshold statement requires a positive-dimensional base and a nef negative log canonical class; its stated coefficient scope is rational boundaries in characteristic zero and real boundaries over $\mathbb{C}$. A Lean formalization is supplied for the family, specifically identified here with the companion on uniform Cartier sections; the provided text does not establish that every claim in the main manuscript is included in that formalization. It does not specify which cases beyond these hypotheses remain open.

Mini glossary

$ϵ$-log-canonical

—

The singularity condition, parameterized by $ϵ$, imposed on the geometric objects in these results.

Klt complement

—

The complement sought in the main manuscript, required to satisfy the singularity condition called klt.

Complement index

—

The integer attached to a complement whose uniform bound is the central conclusion of the main result.

Cartier divisor

—

The type of divisor that the companion selects through a prescribed base point.

Log canonical threshold

—

The singularity threshold of the pulled-back divisor for which the companion gives a uniform lower bound.

Keywords : Klt complements · Fano contractions · Uniform Cartier sections

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Manuscripts in this family