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Math-2535 · mathematics · Algebraic geometry

Bounded denominators in low-dimensional log Calabi–Yau fibrations

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Fix an integer $1\le d\le4$ and a finite set $\Phi\subset[0,1]\cap\mathbb Q$. All varieties in this question are defined over $\mathbb C$. Are there positive integers $p_0\mid p$ and a rational DCC set $\mathcal B\subset[0,1]$, depending only on $d$ and $\Phi$, with the following property?

Let $(X,B_X)$ be a projective log canonical pair of dimension $d$, with boundary coefficients in $\Phi$. Let $f:X\to Z$ be a contraction to a positive-dimensional normal projective base, and let $D$ be a rational Cartier divisor on $Z$ such that \[ K_X+B_X\sim_{\mathbb Q}f^*D. \] Must there be a nonzero rational function $\psi\in\mathbb C(X)^*$ and a rational Cartier divisor $D_Z\sim_{\mathbb Q}D$ satisfying \[ K_X+B_X+\frac1{p_0}\operatorname{div}(\psi)=f^*D_Z, \qquad D_Z=K_Z+B_Z+M_Z, \] where $B_Z$ is effective with coefficients in $\mathcal B$, the generalized pair $(Z,B_Z+M_Z)$ is log canonical and is klt whenever $(X,B_X)$ is klt, and $M_Z$ is the trace of a b-nef moduli b-divisor $\mathbf M$ for which $p\mathbf M$ is b-Cartier?

A contraction here is a surjective projective morphism with $f_*\mathcal O_X=\mathcal O_Z$. The DCC condition means that every decreasing sequence of coefficients eventually stabilizes. Being b-nef and b-Cartier means that on a birational model where $\mathbf M$ descends, its representing divisor is nef and its indicated multiple is Cartier.

Short English statement

Fix $1\le d\le4$ and a finite rational coefficient set $\Phi\subset[0,1]$. Do positive integers $p_0\mid p$ and a rational DCC set $\mathcal B\subset[0,1]$, depending only on $d,\Phi$, give every complex projective log canonical $d$-fold $(X,B_X)$ with coefficients in $\Phi$, and every contraction $f:X\to Z$ to a positive-dimensional normal projective base with $K_X+B_X\sim_{\mathbb Q}f^*D$ for a rational Cartier divisor $D$, an exact presentation $K_X+B_X+p_0^{-1}\operatorname{div}(\psi)=f^*(K_Z+B_Z+M_Z)$? The base divisor must be rationally linearly equivalent to $D$, $B_Z$ must be effective with coefficients in $\mathcal B$, the resulting generalized pair must be log canonical and klt for klt input, and the moduli b-divisor must be b-nef with its $p$-multiple b-Cartier.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2535-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2535

The source treats projective complex log canonical pairs, with a rational Cartier divisor in the pullback hypothesis. With those conventions, its finite coefficient set, dimensions $1\le d\le4$ and positive-dimensional base agree with this question. It asserts the exact rational-function presentation with uniform $p_0\mid p$, effective discriminant in a rational DCC set, and b-nef moduli with $p$ times it b-Cartier. It also retains the generalized lc conclusion and generalized klt conclusion for klt inputs.

Bounded denominators in low-dimensional log Calabi–Yau fibrations

OpenAI's claim

Uniform moduli denominators, trivializing degrees, and effective rounded systems are claimed for the stated log Calabi--Yau fibrations of total dimension at most four. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
Fix a finite $\Phi\subset[0,1]\cap\mathbb Q$ and $1\le d\le4$. For every projective complex log canonical $d$-fold pair $(X,B)$ with coefficients in $\Phi$ and a contraction $f:X\to Z$, $\dim Z>0$, with $K_X+B\sim_{\mathbb Q}f^*D$ for a rational Cartier divisor $D$ on $Z$, OpenAI claims uniform positive integers $p_0\mid p$ and a rational descending-chain-condition set $\mathcal B\subset[0,1]$, depending only on $(d,\Phi)$, such that \[ K_X+B+p_0^{-1}\operatorname{Div}(\psi)=f^*D_Z,\qquad D_Z\sim_{\mathbb Q}D,\quad D_Z=K_Z+B_Z+M_Z \] for some rational function $\psi$. The integer $p_0$ clears $\Phi$, $B_Z\ge0$ has coefficients in $\mathcal B$, and the moduli b-divisor $\mathbf M$ is b-nef with $p\mathbf M$ b-Cartier. The resulting generalized pair is log canonical, and klt when the original pair is klt. This is the denominator result in dimensions at most four, not a bound in unrestricted dimensions.

Outline of the argument

Manuscript 1 (family 034, 2026-09-27): Reduction to curves measures monodromy characters using the index of the entire reduced special fiber. Bounded generalized pairs, smooth-locus topology, Kummer theory, and finitely generated divisor classes provide a common torsion exponent, after which positive base divisors yield effective field-generating systems.

Formal verification

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Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources

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