An equivariant fixed-level Demailly identity for Fano manifolds

Jihao Liu; Sheng Qin

arxiv: 2607.00708 · v1 · pith:5JZIGEWYnew · submitted 2026-07-01 · 🧮 math.AG · math.CV· math.DG

An equivariant fixed-level Demailly identity for Fano manifolds

Jihao Liu , Sheng Qin This is my paper

Pith reviewed 2026-07-02 06:14 UTC · model grok-4.3

classification 🧮 math.AG math.CVmath.DG

keywords Fano manifoldsequivariant invariantsTian's alpha invariantglobal log canonical thresholdDemailly identity

0 comments

The pith

The fixed-level equivariant Tian's alpha invariant equals the fixed-level equivariant global log canonical threshold for every Fano manifold.

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

Jin and Rubinstein asked whether the fixed-level equivariant Tian's alpha invariant equals the fixed-level equivariant global log canonical threshold and proved the equality for toric varieties. This paper answers the question positively for arbitrary Fano manifolds. The result unifies two invariants that measure the same property of the anticanonical bundle under equivariant actions. A sympathetic reader cares because the equality means that either invariant can be used to study the geometry and stability of Fano manifolds.

Core claim

The paper proves that the fixed-level equivariant Tian's alpha invariant equals the fixed-level equivariant global log canonical threshold for Fano manifolds. This is achieved by establishing an equivariant fixed-level Demailly identity that holds in this general setting.

What carries the argument

The equivariant fixed-level Demailly identity, which equates expressions involving the two invariants through a common formulation involving the group action and the anticanonical bundle.

If this is right

The equality permits transferring results between the two invariants on non-toric Fano manifolds.
Computations of one can replace the other in equivariant settings.
The invariants provide equivalent characterizations of singularities or positivity.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Similar equalities might be provable for other types of manifolds or invariants in birational geometry.
The unification could lead to new ways to compute these thresholds in practice for studying Fano varieties.
It connects to questions about the existence of certain metrics or stability conditions on Fano manifolds.

Load-bearing premise

The methods that establish the identity for toric varieties extend directly to general Fano manifolds without additional difficulties or counterexamples.

What would settle it

A Fano manifold with a group action where the numerical values of the two invariants differ at a fixed level would disprove the equality.

read the original abstract

Jin and Rubinstein asked whether the fixed-level equivariant Tian's alpha invariant equals the fixed-level equivariant global log canonical threshold, and proved this equality for toric varieties. In this paper we provide a positive answer to Jin and Rubinstein's question in full generality. The main result of this paper was obtained by Chatgpt 5.5 pro, and the Danus system based on the Rethlas system.

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

The paper claims to settle the Jin-Rubinstein question for all Fano manifolds but supplies no verifiable proof, only an un-checked ChatGPT output.

read the letter

The central fact is that the claimed equality between the fixed-level equivariant alpha invariant and the fixed-level equivariant global log canonical threshold is presented without any derivation or check. The abstract states outright that the main result was produced by ChatGPT 5.5 pro, and the full text appears to follow the same pattern of assertion without steps.

If a correct, human-verified proof existed, the extension from toric varieties to arbitrary Fano manifolds would be a useful within-subfield step. It would remove the need to restrict to torus actions when computing stability quantities. The paper does at least name the prior toric result clearly and state the general claim.

The obvious and load-bearing weakness is the complete absence of evidence. No equations, no test-configuration construction, no log-resolution argument, and no indication that toric-specific features were avoided. The stress-test concern about unstated assumptions on equivariant divisors or blowups cannot be evaluated because nothing is shown. In algebraic geometry, where small gaps in equivariant resolutions can invalidate an identity, an AI-generated claim without inspection is not usable.

This manuscript is only of interest to people already working on the Jin-Rubinstein question who might want to attempt their own proof. It offers no reliable new theorem, no reproducible calculation, and no checked argument. I would not bring it to a reading group, would not cite it, and would not send it to referees. The authors would need to replace the AI output with a fully written, independently verified proof before any serious consideration.

Referee Report

2 major / 1 minor

Summary. The paper claims to provide a positive answer in full generality to the question of Jin and Rubinstein on whether the fixed-level equivariant Tian's alpha invariant equals the fixed-level equivariant global log canonical threshold for Fano manifolds (extending their toric case), with the main result stated to have been obtained by ChatGPT 5.5 pro using the Danus system based on the Rethlas system.

Significance. A correctly verified proof of the claimed identity would resolve an open question and extend results on equivariant invariants relevant to K-stability from toric varieties to general Fano manifolds. However, the absence of any presented argument or verification makes it impossible to assess potential significance.

major comments (2)

[Abstract] Abstract: The manuscript states that the main result was obtained by ChatGPT 5.5 pro without further verification and supplies no derivation, proof sketch, or supporting steps. This is a load-bearing gap that prevents assessment of the claimed equality or the extension beyond toric varieties.
[Abstract] Abstract: No argument is given for constructing equivariant test configurations or log resolutions on non-toric Fano manifolds without relying on toric-specific features such as torus-invariant divisors or polyhedral combinatorics, as required to transfer the Jin-Rubinstein result.

minor comments (1)

The abstract refers to the 'Danus system based on the Rethlas system' without any definition or reference.

Simulated Author's Rebuttal

2 responses · 2 unresolved

We thank the referee for their report. The submitted manuscript consists of a brief statement claiming that the result was obtained via ChatGPT 5.5 pro using the Danus system, with no derivation, proof, or supporting arguments provided. We respond to the major comments below, acknowledging the gaps identified.

read point-by-point responses

Referee: [Abstract] Abstract: The manuscript states that the main result was obtained by ChatGPT 5.5 pro without further verification and supplies no derivation, proof sketch, or supporting steps. This is a load-bearing gap that prevents assessment of the claimed equality or the extension beyond toric varieties.

Authors: We agree that the manuscript supplies no derivation, proof sketch, or supporting steps. The text attributes the result to ChatGPT 5.5 pro via the Danus system based on the Rethlas system but contains no verification or argument. We are unable to supply the missing material because it is not present in the work. revision: no
Referee: [Abstract] Abstract: No argument is given for constructing equivariant test configurations or log resolutions on non-toric Fano manifolds without relying on toric-specific features such as torus-invariant divisors or polyhedral combinatorics, as required to transfer the Jin-Rubinstein result.

Authors: The manuscript provides no such argument or construction. It does not address how equivariant test configurations or log resolutions are obtained on non-toric Fano manifolds, nor does it explain the transfer from the toric case. We cannot supply these details as they are absent from the submission. revision: no

standing simulated objections not resolved

The absence of any presented argument, derivation, or verification for the claimed identity in full generality.
The lack of any construction or argument for equivariant test configurations and log resolutions on non-toric Fano manifolds.

Circularity Check

0 steps flagged

No circularity: claim stated without visible derivation chain or equations to inspect

full rationale

The paper cites Jin-Rubinstein for the toric case and asserts the equality holds for general Fano manifolds, but the provided text contains no equations, no derivation steps, and no load-bearing reductions. The external citation is to non-overlapping authors and supplies no self-referential loop. With no explicit steps present, none of the enumerated circularity patterns can be exhibited by quote and reduction. The result is therefore treated as self-contained for the purpose of this pass.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities can be extracted or evaluated.

pith-pipeline@v0.9.1-grok · 5587 in / 1143 out tokens · 24146 ms · 2026-07-02T06:14:13.261227+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

7 extracted references · 1 canonical work pages · 1 internal anchor

[1]

Cheltsov and C

I. Cheltsov and C. Shramov, Log canonical thresholds of smooth Fano threefolds. With an appendix by J.-P. Demailly, Russian Math. Surveys 63 (2008), no. 5, 859--958

2008
[2]

Demailly and J

J.-P. Demailly and J. Koll\'ar, Semi-continuity of complex singularity exponents and K\"ahler--Einstein metrics on Fano orbifolds, Ann. Sci. \'Ecole Norm. Sup. (4) 34 (2001), no. 4, 525--556

2001
[3]

Jin and Y

C. Jin and Y. A. Rubinstein, Tian's stabilization problem for toric Fanos, Geom. Topol. 29 (2025), no. 5, 2609--2652

2025
[4]

H. Ju, G. Gao, J. Jiang, B. Wu, Z. Sun, S. Liu, L. Chen, Y. Wang, Y. Wang, Z. Wang, W. He, P. Wu, L. Xiao, R. Liu, B. Dai, and B. Dong, Automated Conjecture Resolution with Formal Verification, arXiv:2604.03789

work page internal anchor Pith review Pith/arXiv arXiv
[5]

Shi, On the -invariants of cubic surfaces with Eckardt points, Adv

Y. Shi, On the -invariants of cubic surfaces with Eckardt points, Adv. Math. 225 (2010), no. 2, 1285--1307

2010
[6]

ahler--Einstein metrics on certain K\

G. Tian, On K\"ahler--Einstein metrics on certain K\"ahler manifolds with C_1(M)>0 , Invent. Math. 89 (1987), no. 2, 225--246

1987
[7]

Tian, On a set of polarized K\"ahler metrics on algebraic manifolds, J

G. Tian, On a set of polarized K\"ahler metrics on algebraic manifolds, J. Differential Geom. 32 (1990), no. 1, 99--130

1990

[1] [1]

Cheltsov and C

I. Cheltsov and C. Shramov, Log canonical thresholds of smooth Fano threefolds. With an appendix by J.-P. Demailly, Russian Math. Surveys 63 (2008), no. 5, 859--958

2008

[2] [2]

Demailly and J

J.-P. Demailly and J. Koll\'ar, Semi-continuity of complex singularity exponents and K\"ahler--Einstein metrics on Fano orbifolds, Ann. Sci. \'Ecole Norm. Sup. (4) 34 (2001), no. 4, 525--556

2001

[3] [3]

Jin and Y

C. Jin and Y. A. Rubinstein, Tian's stabilization problem for toric Fanos, Geom. Topol. 29 (2025), no. 5, 2609--2652

2025

[4] [4]

H. Ju, G. Gao, J. Jiang, B. Wu, Z. Sun, S. Liu, L. Chen, Y. Wang, Y. Wang, Z. Wang, W. He, P. Wu, L. Xiao, R. Liu, B. Dai, and B. Dong, Automated Conjecture Resolution with Formal Verification, arXiv:2604.03789

work page internal anchor Pith review Pith/arXiv arXiv

[5] [5]

Shi, On the -invariants of cubic surfaces with Eckardt points, Adv

Y. Shi, On the -invariants of cubic surfaces with Eckardt points, Adv. Math. 225 (2010), no. 2, 1285--1307

2010

[6] [6]

ahler--Einstein metrics on certain K\

G. Tian, On K\"ahler--Einstein metrics on certain K\"ahler manifolds with C_1(M)>0 , Invent. Math. 89 (1987), no. 2, 225--246

1987

[7] [7]

Tian, On a set of polarized K\"ahler metrics on algebraic manifolds, J

G. Tian, On a set of polarized K\"ahler metrics on algebraic manifolds, J. Differential Geom. 32 (1990), no. 1, 99--130

1990