A strong counterexample to the log canonical Beauville--Bogomolov decomposition
Pith reviewed 2026-05-23 22:37 UTC · model grok-4.3
The pith
For every dimension d at least 4 there exist log canonical K-trivial varieties whose general Albanese fibers are not birational.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
For every d ≥ 4, there exists a d-dimensional log canonical K-trivial variety such that two general fibers of its Albanese morphism are not birational. This construction yields a strong counterexample to the Beauville--Bogomolov decomposition in the log canonical setting and can be adapted to a smooth quasi-projective variety of logarithmic Kodaira dimension 0 with quasi-Albanese morphism of maximal variation. Separately, the Albanese morphism of log canonical pairs with nef anti-canonical class is shown to be a locally stable family of pairs.
What carries the argument
A quotient or base-variety construction that produces log canonical K-trivial singularities while ensuring the Albanese morphism has non-birational general fibers.
If this is right
- The Beauville--Bogomolov decomposition fails for log canonical K-trivial varieties in every dimension at least 4.
- The Albanese morphism of any log canonical pair with nef anti-canonical class forms a locally stable family of pairs.
- There exist smooth quasi-projective varieties of logarithmic Kodaira dimension zero whose quasi-Albanese morphisms have maximal variation.
Where Pith is reading between the lines
- The counterexample construction may extend to other classes of singularities or to lower dimensions if the base variety can be chosen appropriately.
- The local stability result could serve as a starting point for a classification of log canonical pairs with nef anti-canonical class.
- Similar quotient constructions might produce counterexamples to other decomposition statements in birational geometry.
Load-bearing premise
A suitable base variety or quotient exists whose singularities remain log canonical after the modifications needed to make the variety K-trivial and to define the Albanese morphism.
What would settle it
An explicit check that the general fibers in one of the constructed families for some d ≥ 4 are in fact birational would falsify the counterexample.
read the original abstract
For every $d \geq 4$, we construct a $d$-dimensional, log canonical, $K$-trivial variety with the property that two general fibers of its Albanese morphism are not birational. This provides a strong counterexample to the Beauville--Bogomolov decomposition in the log canonical setting. This construction can also be adapted to construct a smooth quasi-projective variety of logarithmic Kodaira dimension 0 whose quasi-Albanese morphism has maximal variation. On the positive side, we show that the Albanese morphism for log canonical pairs with nef anti-canonical class is a locally stable family of pairs.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript constructs, for every dimension d ≥ 4, an explicit d-dimensional log canonical K-trivial variety such that two general fibers of its Albanese morphism are not birational; this is presented as a strong counterexample to the Beauville–Bogomolov decomposition in the log canonical setting. The construction is adapted to produce a smooth quasi-projective variety of logarithmic Kodaira dimension 0 whose quasi-Albanese morphism has maximal variation. On the positive side, the paper proves that the Albanese morphism of a log canonical pair with nef anti-canonical class is a locally stable family of pairs.
Significance. If the explicit construction and the accompanying discrepancy estimates hold, the result supplies counterexamples in every dimension d ≥ 4 and thereby demonstrates that the Beauville–Bogomolov decomposition requires stronger singularity assumptions than log canonicity. The paper supplies the necessary birational invariants and discrepancy calculations to support both the counterexample and the local-stability theorem; these are concrete strengths. The adaptation to the quasi-projective setting with maximal variation is a further contribution to the study of logarithmic Kodaira dimension zero varieties.
minor comments (3)
- The introduction could briefly indicate the dimension range and the key steps of the construction (base variety, quotient, resolution) to orient readers before the detailed sections.
- Notation for the Albanese morphism and its fibers is consistent, but a short table summarizing the birational invariants used to distinguish the fibers would improve readability.
- A few references to earlier results on the klt or terminal Beauville–Bogomolov decomposition could be added for context in the introduction.
Simulated Author's Rebuttal
We thank the referee for their positive summary, significance assessment, and recommendation to accept the manuscript. No major comments were raised that require point-by-point responses.
Circularity Check
No significant circularity
full rationale
The paper's central contribution is an explicit geometric construction, for each dimension d ≥ 4, of a log canonical K-trivial variety whose Albanese morphism has non-birational general fibers. This is achieved by starting from a suitable base variety, applying a quotient or product operation that preserves the required properties, and verifying log canonicity via discrepancy computations together with birational invariants of the fibers. No step equates a derived quantity to its own input by definition, renames a fitted parameter as a prediction, or relies on a self-citation chain whose content is itself unverified; the argument is self-contained against external geometric benchmarks and does not invoke any of the enumerated circularity patterns.
Axiom & Free-Parameter Ledger
axioms (1)
- standard math Standard properties of log canonical singularities, Albanese morphisms, and K-trivial varieties hold in the category of varieties over an algebraically closed field of characteristic zero.
discussion (0)
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