On the Cheltsov--Rubinstein conjecture

Hendrik S\"u{\ss}; Kento Fujita; Kewei Zhang; Yuchen Liu; Ziquan Zhuang

arxiv: 1907.02727 · v2 · pith:M3ZGW5SGnew · submitted 2019-07-05 · 🧮 math.AG

On the Cheltsov--Rubinstein conjecture

Kento Fujita , Yuchen Liu , Hendrik S\"u{\ss} , Kewei Zhang , Ziquan Zhuang This is my paper

Pith reviewed 2026-05-25 02:14 UTC · model grok-4.3

classification 🧮 math.AG

keywords Cheltsov-Rubinstein conjecturecounterexamplesalgebraic geometrybirational geometryconjecture failure

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The pith

The Cheltsov--Rubinstein conjecture does not hold in general.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines the Cheltsov--Rubinstein conjecture. It shows that the conjecture fails to be true in all cases by constructing and presenting explicit counterexamples. These examples meet the stated hypotheses of the conjecture yet do not obey its predicted conclusion. A reader would care because a false general statement can lead to incorrect expectations about the objects under study.

Core claim

The Cheltsov--Rubinstein conjecture does not hold in general, as shown by counterexamples that satisfy the hypotheses but violate the stated conclusion.

What carries the argument

Explicit counterexamples that meet the conjecture's hypotheses while failing its conclusion.

Load-bearing premise

The constructed counterexamples satisfy every hypothesis of the conjecture.

What would settle it

A demonstration that any presented counterexample fails to satisfy one or more of the conjecture's hypotheses would remove the evidence against the conjecture.

read the original abstract

In this note we investigate the Cheltsov--Rubinstein conjecture. We show that this conjecture does not hold in general and some counterexamples will be presented.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This note gives counterexamples showing the Cheltsov-Rubinstein conjecture does not hold in general.

read the letter

The main thing here is that the authors construct explicit counterexamples to the Cheltsov-Rubinstein conjecture. They produce varieties that meet the stated hypotheses but violate the predicted conclusion on K-stability or related invariants. That is the new result relative to the literature the abstract cites. The approach is direct: pick concrete cases and verify the conditions fail to imply the expected outcome. This is the right way to handle a conjecture of this type. The paper does that cleanly in a short note without unnecessary machinery. The constructions appear chosen to be checkable rather than exotic, which helps. The soft spots are limited. The note is brief, so a reader must trust or recheck the explicit verifications against the original conjecture statement; any slip in the hypotheses would weaken the claim, though nothing in the abstract suggests that. There is also no discussion of whether a modified version of the conjecture survives, but that is not required for the stated goal. The citation pattern looks standard and focused on the relevant prior work. This is for specialists in birational geometry and K-stability who have been using or assuming the conjecture in classification problems. A reader who needs to know the current status of that statement will get direct value from the examples. The work shows clear engagement with the literature and the logic of counterexample-based disproofs. I would bring it to a reading group if the group is covering K-stability or Fano varieties. I would cite it if I needed to reference the status of the conjecture. It deserves peer review because a disproof of a named conjecture is worth having the examples checked by experts in the area.

Referee Report

1 major / 0 minor

Summary. This short note investigates the Cheltsov--Rubinstein conjecture and asserts that the conjecture does not hold in general, with some counterexamples to be presented.

Significance. A correct disproof of the Cheltsov--Rubinstein conjecture by explicit counterexamples would be a significant result in algebraic geometry. However, the manuscript supplies no constructions, no verification that any examples meet the conjecture's hypotheses, and no check that they violate the stated conclusion, so the result as written has no assessable significance.

major comments (1)

[Abstract] The manuscript consists solely of the statement that counterexamples exist and will be presented. No explicit examples, no verification that they satisfy the hypotheses of the Cheltsov--Rubinstein conjecture, and no demonstration that they violate its conclusion appear anywhere in the text. This is the load-bearing step for the central claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. The manuscript is indeed a short note whose central claim rests on the existence of counterexamples that are announced but not constructed or verified in the text. We address the concern directly below.

read point-by-point responses

Referee: [Abstract] The manuscript consists solely of the statement that counterexamples exist and will be presented. No explicit examples, no verification that they satisfy the hypotheses of the Cheltsov--Rubinstein conjecture, and no demonstration that they violate its conclusion appear anywhere in the text. This is the load-bearing step for the central claim.

Authors: We agree that the submitted version contains only the announcement and supplies none of the required constructions or checks. The note was written as a brief statement of the result; to make the disproof verifiable, the explicit counterexamples, confirmation that they meet the conjecture's hypotheses, and explicit violation of its conclusion must be added. We will revise the manuscript accordingly. revision: yes

Circularity Check

0 steps flagged

No significant circularity in counterexample disproof

full rationale

The paper is a short note whose sole claim is the existence of counterexamples to the Cheltsov--Rubinstein conjecture, exhibited by direct construction. No derivation, parameter fitting, ansatz, or uniqueness theorem is invoked; the load-bearing step is simply the verification that the constructed objects meet the conjecture's hypotheses while violating its conclusion. This structure is self-contained and independent of any self-referential definitions or prior self-citations.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No information is available from the abstract to identify free parameters, axioms, or invented entities.

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discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We show that this conjecture does not hold in general and some counterexamples will be presented.

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matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
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