Bounded klt complements and uniformly mild hypersurfaces

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Family 066 · Algebraic and complex geometry
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-12

1. The question

Suppose a geometric space has singularities, but none are too severe. Can one choose an equation through any prescribed point without creating much worse singularities? And can one balance the canonical divisor by an effective boundary using a denominator bounded uniformly across a whole class of spaces?

These sound like different questions. One concerns choosing a local hypersurface; the other concerns constructing a global-looking divisor near a fiber. Earlier work shows that, for suitable Fano-type maps, the two are closely connected.

The two manuscripts in this family claim uniform answers in every dimension. The relevant bounds depend on dimension and an initial singularity gap, with a finite coefficient set additionally needed for the complement index. The geometry and the chosen equation may vary freely within the stated hypotheses.

2. Background and essential definitions

Measuring singularities

A pair $(X,B)$ consists of a normal algebraic variety and an effective boundary divisor $B$, a weighted sum of codimension-one subvarieties. Assume $K_X+B$ is rationally Cartier: an integer multiple is locally given by one equation, or equivalently defines a line bundle. The canonical divisor $K_X$ records top-degree differential forms.

To examine a singularity, pass to a smooth birational model $\pi:Y\to X$. For a prime divisor $E$ on such a model, its log discrepancy is

$$A_{X,B}(E)=1+\operatorname{ord}_E\bigl(K_Y-\pi^*(K_X+B)\bigr).$$

This measures how differential forms and the boundary transform near that divisor. All such divisors, including those visible only after further modifications, must be considered. The pair is log canonical, abbreviated lc, if every discrepancy is nonnegative; Kawamata log terminal, abbreviated klt, if every discrepancy is positive; and $\epsilon$-lc if every discrepancy is at least a fixed $\epsilon>0$.

Thus klt is stricter than lc, and $\epsilon$-lc specifies a quantitative margin. The margin is imposed on prime-divisor orders in their natural normalization. Multiplying an order function by an arbitrarily small real number also scales its discrepancy, so unrestricted rescaling cannot be used to test this condition.

Fano contractions and complements

A contraction $f:X\to Z$ is a projective surjective map of normal varieties with $f_*\mathcal O_X=\mathcal O_Z$. A Fano contraction has $-K_X$ ample relative to the base. Relative ampleness concerns positivity in the directions of the fibers.

More generally, $X$ is of Fano type over $Z$ if some effective rational boundary $C$ makes $(X,C)$ klt and $-(K_X+C)$ relatively ample. This auxiliary boundary need not be the boundary $B$ appearing in the theorem. A divisor is relatively nef when its intersection with every contracted curve is nonnegative.

A monotone klt complement replaces $B$ by a larger effective rational boundary $B^+\geq B$ such that the resulting pair is klt and

$$N(K_X+B^+)\sim0$$

near the chosen fiber. Here the displayed divisor is integral Cartier and linearly trivial. Linear triviality is stronger than having zero intersection with curves: the corresponding line bundle must actually be trivial. The integer $N$ is the complement index.

Thresholds explain the local question

For an effective Cartier divisor $D$ on the base, the log canonical threshold of its pullback is the largest allowable coefficient in $B+t f^*D$. It is controlled by ratios

$$\frac{A_{X,B}(E)}{\operatorname{ord}_E(f^*D)}$$

over divisors with positive denominator. The initial discrepancy bound controls the numerator but says nothing immediately about the denominator. On a smooth curve, a divisor consisting of a point with multiplicity $m$ has threshold $1/m$. The theorem must therefore choose a good equation; it cannot make every equation good.

3. Development history and previous partial results

Shokurov introduced complements in the study of threefold flips and developed their boundedness systematically in his work on surfaces. Prokhorov and Shokurov developed the higher-dimensional program and methods passing between global and local complements.

Birkar’s bounded-complement theorem established bounded lc complements for Fano-type pairs with suitable coefficient sets in arbitrary dimension. This is a major foundation for the present manuscripts. An lc complement may have a divisor of discrepancy zero, however. Simply reducing its coefficients can destroy the exact linear equivalence that made it a complement, so obtaining a bounded klt complement is a separate problem.

Partial results controlled that stronger requirement. Birkar’s work on Fano fibrations gives bounded klt complements over curves. Toric geometry and weighted blowups provide special settings where orders of equations can be controlled more explicitly; the manuscripts discuss results of Ambro and of Sankaran–Santos in these directions.

The decisive organizing result is Bingyi Chen’s equivalence theorem. It links bounded klt complements to the existence of a uniformly mild Cartier hypersurface through a prescribed point. Chen proves the local surface case and consequently the complement conjecture for bases of dimension at most two. This concerns the dimension of the base, not merely the dimension of the total space.

The new manuscripts use that reduction while proposing different arguments for its remaining all-dimensional input. They also build on normalized-volume theory for singularities, which constructs canonical degenerations by optimizing over orders of vanishing. Existing optimization theorems concern ordinary klt pairs; both manuscripts identify additional work needed to impose their special constraints.

4. What the new manuscripts claim

M1: bounded klt complements

Fix a positive integer $d$ and a positive rational $\epsilon$. M1 claims an integer $N=N(d,\epsilon)$ for every $\epsilon$-lc Fano contraction $f:X\to Z$ of total dimension $d$, over an algebraically closed field of characteristic zero.

For every closed point $z\in Z$, there are an effective rational boundary $B$ and a Zariski-open neighborhood $U$ of $z$ such that $(X,B)$ is klt over $U$ and

$$N(K_X+B)|_{f^{-1}(U)}\sim0.$$

Neither relative Picard number one nor positive-dimensional general fibers is required. The boundary and neighborhood depend on the contraction and point; the integer does not.

Over $\mathbb C$, M1 also claims monotone klt complements for $\epsilon$-lc pairs $(X,\Delta)$ with coefficients in a fixed finite rational set $I$, provided $X$ is of Fano type over the quasi-projective base and $-(K_X+\Delta)$ is relatively nef. Here the uniform index depends on $d,\epsilon,I$. This is the finite-rational-coefficient form of the bounded-klt-complement conjecture.

M2: a uniformly mild Cartier section

M2 claims that for every $d$ and $\epsilon>0$, there is a positive number $\tau(d,\epsilon)$ with the following property. Let $(X,B)$ be an $\epsilon$-lc rational pair of dimension $d$, and let $f:X\to Z$ be a Fano-type contraction of normal quasi-projective varieties in characteristic zero, with positive-dimensional base and $-(K_X+B)$ relatively nef.

For each closed point $z$, one can find an effective nonzero Cartier divisor $D$ through $z$, defined on a neighborhood $U$, such that

$$\bigl(f^{-1}(U),B|_{f^{-1}(U)}+\tau f^*D\bigr) \quad\text{is log canonical}.$$

The constant is independent of boundary denominators, Cartier indices and polarizations. Over $\mathbb C$, the manuscript extends this conclusion to effective real boundaries with real-Cartier log canonical divisor.

M2 separately proves a bounded klt complement statement for birational Fano-type contractions with zero boundary and relatively nef $-K_X$. Through Chen’s equivalence, this supplies the Cartier-section claim and the complex finite-coefficient complement consequence. The manuscripts present their principal arguments as independent approaches, rather than one paper merely invoking the other.

5. Why the advance matters

Importance: 5/5. The claims address a central uniformity problem in the study of Fano varieties and singularities. Complements let geometry with positive anticanonical direction be replaced by a controlled log-Calabi–Yau pair, where the adjusted canonical divisor becomes torsion. Uniform indices make that replacement useful across families.

The local section theorem is equally concrete: a prescribed base point admits an equation whose pullback remains tolerably singular along the whole nearby fiber. It is stronger than controlling thresholds only at generic points of divisors on the base.

The rating reflects the breadth of these claimed all-dimensional results and their relationship with existing boundedness theory. It measures significance if correct, not proof confidence.

6. The main idea

Turn failure of uniformity into concentration

M1 reduces the problem to a lower bound for the threshold of the pulled-back maximal ideal of a base point in the birational, zero-boundary setting. If no such bound existed, one could find a sequence in which orders of that ideal become arbitrarily large relative to discrepancies.

Anticanonical sections are collected into a graded ring, whose spectrum is an affine cone. A bounded lc complement supplies an initial system of fixed degree. The proposed constrained optimization chooses an order function on the cone that both minimizes normalized volume and lies on a prescribed log-canonical boundary condition.

A crucial claimed step realizes that constrained minimizer as the actual minimizer for one small klt perturbation. This enables the established stable-degeneration machinery to produce a finitely generated graded model without losing the concentration information.

Count sections, then recover a genuine divisor

M1 organizes independent directions in the resulting function fields according to the cost of producing sections. Upper estimates limit how many sections can occur at a given cost; independent monomials provide competing lower estimates. Auxiliary reductions to positive characteristic use Cartier trace, an operation on differential forms associated with the Frobenius map, to construct controlled sections.

The endpoint must be more precise than “some real valuation has small discrepancy.” Such smallness can be manufactured by rescaling. The manuscript therefore tracks independent coordinates and denominators until a bounded multiple of the constructed valuation is a positive integral multiple of a prime-divisor order. Its discrepancy then contradicts the original $\epsilon$-lc bound in the required normalization.

A second route through characters

M2 starts from cyclic covers and sections transforming under specified characters, meaning scalar multiplication rules for the finite cyclic group. Failure of bounded complements forces every section of a fixed character degree to have a large order.

Dividing such sections by powers of a distinguished function produces invariant rational functions. Requiring both a function and its inverse to have nonnegative order pins their order exactly to zero. The argument repeatedly preserves these equalities while adding algebraically independent functions.

Each successful addition uses one of the finitely many dimensions available in the function field. If the process cannot add another independent function, the remaining character direction is claimed to produce a prime divisor of too-small discrepancy on the quotient. Thus dimension bounds one alternative, while the singularity gap rules out the other. Chen’s equivalence then translates this obstruction into the stated geometric conclusions.

7. Scope, limitations, and open questions

These are local statements over a neighborhood of each chosen base point. They do not claim one boundary or one hypersurface works simultaneously over an entire arbitrary base without shrinking.

The complement output is klt, but is not asserted to preserve the original $\epsilon$-lc bound. The stronger demand for quantitatively bounded discrepancies in the complement remains distinct. Likewise, the finite rational coefficient set enters the complement index even though the Cartier-section coefficient has no such dependence.

The bounds are existence statements, not explicit numerical formulas. The use of positive characteristic in M1 is an auxiliary method inside a characteristic-zero theorem; it does not establish the same result for varieties over arbitrary positive-characteristic fields.

The real-boundary extension is stated over $\mathbb C$. Rational-boundary results have the broader characteristic-zero scope specified above. Preserving these different fields, boundary types and uniformity dependencies is essential to understanding what each manuscript contributes.

8. Sources and further reading

Chen’s paper provides the closest formulation of the conjectures, their equivalence and the surface-base result. Birkar’s bounded lc complements and his work on Fano fibrations explain the main earlier boundedness inputs. For the optimization background, see Xu–Zhuang on stable degenerations.

Source manuscripts

• M1. OpenAI, Bounded klt complements for Fano contractions (September 25, 2026).
• M2. OpenAI, Uniform Cartier sections for Fano type contractions (September 25, 2026).

Sources consulted on 12 October 2026 at repository revision fd4aeeb. Download the manuscript citations.