arxiv: 2601.02901 · v2 · submitted 2026-01-06 · 🧮 math.DG

The 2-systole on compact K\"ahler surfaces with positive scalar curvature

Zehao Sha This is my paper

Pith reviewed 2026-05-16 17:18 UTC · model grok-4.3

classification 🧮 math.DG MSC 53C5553C21

keywords Kähler surfacespositive scalar curvature2-systolesystolic inequalityFubini-Study metricruled surfacesEnriques-Kodaira classification

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The pith

Any compact Kähler surface with positive scalar curvature satisfies min S(ω) · sys₂(ω) ≤ 12π, with equality only for the projective plane with Fubini-Study metric.

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

This paper proves a sharp inequality bounding the product of the minimum scalar curvature and the two-dimensional systole on compact Kähler surfaces that have everywhere positive scalar curvature. The bound equals 12π and is attained precisely on the complex projective plane equipped with its Fubini-Study Kähler form. The argument relies on the complete list of such surfaces from the Enriques-Kodaira classification to compute the best constant case by case. An alternative proof for ruled surfaces uses an adaptation of level-set techniques to the holomorphic fibration structure.

Core claim

For any compact Kähler surface (X, ω) with positive scalar curvature, min_X S(ω) · sys₂(ω) ≤ 12π, with equality if and only if X = ℙ² and ω is the Fubini-Study metric. Using the classification of positive scalar curvature Kähler surfaces, the optimal constant is determined in each case, and rigid models are described. For non-rational ruled surfaces an independent proof adapts Stern's level set method to the holomorphic fibration.

What carries the argument

The Enriques-Kodaira classification of compact Kähler surfaces with positive scalar curvature, which allows case-by-case determination of the optimal constant in the inequality min S · sys₂ ≤ 12π.

Load-bearing premise

The classification of compact Kähler surfaces with positive scalar curvature is complete and exhaustive.

What would settle it

Discovery of a compact Kähler surface with positive scalar curvature where min S(ω) · sys₂(ω) exceeds 12π, or where equality holds on a surface other than ℙ².

read the original abstract

We study the 2-systole on compact K\"ahler surfaces of positive scalar curvature. For any such surface $(X,\omega)$, we prove the sharp estimate $\min_X S(\omega)\cdot\operatorname{sys}_2(\omega)\le 12\pi$, with equality if and only if $X=\mathbb{P}^2$ and $\omega$ is the Fubini-Study metric. Using the classification of positive scalar curvature K\"ahler surfaces, we determine the optimal constant in each case and describe the corresponding rigid models. When $X$ is a non-rational ruled surface, we also give an independent analytic proof, adapting Stern's level set method to the holomorphic fibration in K\"ahler setting.

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 proves a sharp 12π bound on min scalar curvature times 2-systole for compact Kähler surfaces with positive scalar curvature, with equality only on P² with Fubini-Study, by combining classification with an adapted level-set argument.

read the letter

The main result is a clean sharp inequality: for any compact Kähler surface with positive scalar curvature, min S(ω) times the 2-systole is at most 12π, and equality holds precisely when the surface is P² equipped with the Fubini-Study metric. That constant and the equality case are the concrete new output. The paper also works out the optimal constant case by case using the known classification of such surfaces, and for non-rational ruled surfaces it supplies an independent analytic proof by adapting Stern’s level-set technique to the holomorphic fibration. That adaptation looks like the genuinely new technical step; it avoids the classification for those surfaces and still recovers the bound. The rest of the argument is essentially a careful application of the classification to pin down the extremal models. The soft spot is exactly what the stress-test note flags: the global sharpness claim rests on the classification being exhaustive and on the extremal metrics having been correctly identified in each class. If the classification misses a surface or an extremal metric, the optimality statement fails even if the inequality itself holds. The analytic part for ruled surfaces stands on its own, but the paper’s headline optimality statement does not. The math looks formally grounded and the citation pattern is standard for this corner of four-dimensional geometry. This is the kind of paper that belongs in a specialist journal on Kähler geometry or systolic inequalities; a reader working on curvature-topology relations in dimension four would find the bound and the equality case useful as a reference. It is worth sending to referees because the bound is sharp, the method is explicit, and the only real question is whether the classification step is airtight.

Referee Report

2 major / 2 minor

Summary. The manuscript proves that for any compact Kähler surface (X, ω) with positive scalar curvature, min_X S(ω) · sys₂(ω) ≤ 12π, with equality if and only if X = ℙ² and ω is the Fubini-Study metric. The argument proceeds by case analysis on the classification of such surfaces to identify the optimal constant in each class, together with an independent analytic proof for non-rational ruled surfaces that adapts Stern’s level-set technique to the holomorphic fibration.

Significance. If correct, the result supplies a sharp systolic inequality that links the minimum scalar curvature to the 2-systole on PSC Kähler surfaces, with a clean rigidity statement. The combination of exhaustive case analysis and a self-contained analytic argument for ruled surfaces strengthens the conclusion and provides explicit extremal models.

major comments (2)

[Main theorem and case analysis] The sharpness claim (12π is optimal in every case and equality holds only for (ℙ², ω_FS)) rests on the completeness of the invoked classification of compact Kähler surfaces admitting positive-scalar-curvature metrics. The manuscript must cite the precise classification theorem (including the reference) and verify that every class has been checked for possible larger constants.
[Analytic proof for non-rational ruled surfaces] In the independent level-set argument for non-rational ruled surfaces, the adaptation of Stern’s method requires explicit error estimates and a clear treatment of equality cases to confirm that the bound min S · sys₂ ≤ 12π cannot be improved within this class.

minor comments (2)

[Introduction] The notation sys₂(ω) and S(ω) should be defined at first appearance rather than relying solely on the abstract.
[Case analysis] A short table summarizing the optimal constant for each surface class would improve readability of the case analysis.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major point below and have revised the paper accordingly to strengthen the presentation.

read point-by-point responses

Referee: [Main theorem and case analysis] The sharpness claim (12π is optimal in every case and equality holds only for (ℙ², ω_FS)) rests on the completeness of the invoked classification of compact Kähler surfaces admitting positive-scalar-curvature metrics. The manuscript must cite the precise classification theorem (including the reference) and verify that every class has been checked for possible larger constants.

Authors: We agree that an explicit citation and class-by-class verification are required to make the sharpness statement fully rigorous. In the revised manuscript we now cite the precise classification theorem for compact Kähler surfaces admitting positive-scalar-curvature metrics and have added a dedicated subsection that enumerates each class, confirms that the constant 12π is optimal throughout, and identifies the equality cases. No larger constants appear in any class. revision: yes
Referee: [Analytic proof for non-rational ruled surfaces] In the independent level-set argument for non-rational ruled surfaces, the adaptation of Stern’s method requires explicit error estimates and a clear treatment of equality cases to confirm that the bound min S · sys₂ ≤ 12π cannot be improved within this class.

Authors: We thank the referee for this observation. The original analytic argument adapted Stern’s level-set technique to the holomorphic fibration but did not supply fully explicit error estimates or a detailed equality-case analysis. The revised version now includes the missing error bounds (with explicit dependence on the fibration data) and a separate paragraph treating equality cases, showing that the bound cannot be improved inside the class of non-rational ruled surfaces. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external classification theorem and independent adaptation of prior analytic technique

full rationale

The paper establishes the bound min_X S(ω)·sys₂(ω)≤12π via case analysis on the classification of compact Kähler surfaces with positive scalar curvature (an external result) and, separately, via an independent analytic argument adapting Stern's level-set method to the holomorphic fibration for non-rational ruled surfaces. No equation or step reduces the claimed inequality or the sharpness statement to a quantity defined by fitting parameters inside the paper, nor does any load-bearing premise collapse to a self-citation whose content is unverified. The classification is treated as prior independent knowledge that is externally falsifiable; the adaptation of Stern's method is likewise external. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the external classification theorem for positive-scalar-curvature Kähler surfaces and on the validity of the adapted level-set argument; no free parameters are introduced and no new entities are postulated.

axioms (1)

domain assumption Complete classification of compact Kähler surfaces with positive scalar curvature
Invoked to compute the optimal constant and rigid models for each topological type.

pith-pipeline@v0.9.0 · 5417 in / 1292 out tokens · 37489 ms · 2026-05-16T17:18:22.901979+00:00 · methodology

discussion (0)

Forward citations

Cited by 2 Pith papers

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Stable 2-systole bounds in positive scalar curvature
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The stable 2-systole is uniformly bounded above for all metrics with scalar curvature ≥1 on closed spin 2-essential manifolds.