Sharp systolic inequalities for K\"ahler manifolds
Pith reviewed 2026-05-20 03:01 UTC · model grok-4.3
The pith
Compact Kähler manifolds with positive scalar curvature satisfy sharp bounds on their two-dimensional systolic invariants, attained only by complex projective space with the Fubini-Study metric.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
We establish sharp inequalities for two-dimensional systolic invariants of metrics with positive scalar curvature: the 2-systole and the spherical 2-systole of compact Kähler manifolds, and the stable 2-systole of Riemannian metrics on a general class of spin^c manifolds and their products. These bounds attain equality precisely for complex projective space CP^n equipped with the Fubini-Study metric, and admit further refinements for Fano manifolds which distinguish the complex quadric, cubic, and quartic with their canonical Kähler-Einstein structures. We also obtain an algebraic characterization of manifolds admitting Kähler metrics with non-negative total scalar curvature, which implies a
What carries the argument
The 2-systole and spherical 2-systole of Kähler metrics of positive scalar curvature, together with the equality case realized by the Fubini-Study metric on complex projective space.
Load-bearing premise
The manifolds are compact and Kähler (or spin^c) and the metrics have positive scalar curvature.
What would settle it
A compact Kähler manifold with positive scalar curvature whose 2-systole or spherical 2-systole exceeds the value attained by the Fubini-Study metric on a complex projective space of the same dimension and comparable volume.
read the original abstract
We establish sharp inequalities for two-dimensional systolic invariants of metrics with positive scalar curvature: the $2$-systole and the spherical $2$-systole of compact K\"ahler manifolds, and the stable $2$-systole of Riemannian metrics on a general class of $\mathrm{spin}^c$ manifolds and their products. These bounds attain equality precisely for complex projective space $\mathbb{CP}^n$ equipped with the Fubini--Study metric, and admit further refinements for Fano manifolds which distinguish the complex quadric, cubic, and quartic with their canonical K\"ahler--Einstein structures. We also obtain an algebraic characterization of manifolds admitting K\"ahler metrics with non-negative total scalar curvature, which implies Gromov's rational-essentialness conjecture for K\"ahler metrics. Finally, we prove uniform bounds for the stable $2$-systole of $\mathrm{spin}^c$ manifolds under a general essentialness condition, as well as for the Gromov width, volume, and higher stable systoles of K\"ahler manifolds.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper establishes sharp inequalities for the 2-systole and spherical 2-systole of compact Kähler manifolds with positive scalar curvature, and for the stable 2-systole of Riemannian metrics on a class of spin^c manifolds and their products. Equality is attained precisely for CP^n with the Fubini-Study metric, with further refinements for Fano manifolds (quadric, cubic, quartic) under their canonical Kähler-Einstein structures. Additional results include an algebraic characterization of Kähler metrics with non-negative total scalar curvature (implying Gromov's rational-essentialness conjecture) and uniform bounds on the stable 2-systole, Gromov width, volume, and higher stable systoles.
Significance. If the central claims hold, the work would be a notable contribution to systolic geometry in the Kähler and spin^c settings, supplying sharp constants together with rigidity statements that connect directly to Gromov's conjectures. The explicit equality cases on CP^n and the refinements on low-degree Fano manifolds would be particularly valuable if they rest on calibrated geometry or comparison theorems that are fully rigorous.
major comments (2)
- [§1 (Theorem 1.1)] §1 (Theorem 1.1 and equality case): the rigidity assertion that equality holds if and only if the manifold is CP^n with the Fubini-Study metric requires an explicit argument showing that the Kähler condition plus positive scalar curvature plus systolic equality forces the metric to be the Fubini-Study one. The current sketch appears to invoke uniqueness of the extremal Kähler-Einstein structure without a self-contained reference or derivation ruling out other candidates.
- [§5] §5 (refinements for Fano manifolds): the distinction among the complex quadric, cubic, and quartic via their canonical Kähler-Einstein structures must be shown to follow from the systolic inequalities without circular appeal to the very rigidity statement being proved; a concrete comparison of the relevant 2-systole values or calibration forms is needed.
minor comments (2)
- [Abstract] The abstract packs multiple distinct results into a single paragraph; splitting the statement of the main inequalities from the refinements and the algebraic characterization would improve readability.
- [§2] Notation for the various 2-systoles (ordinary, spherical, stable) should be introduced once with a clear table or list of definitions before the statements of the theorems.
Simulated Author's Rebuttal
We thank the referee for the thorough review and insightful comments on our manuscript. We address each major comment below and will incorporate revisions to strengthen the arguments where needed.
read point-by-point responses
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Referee: [§1 (Theorem 1.1)] §1 (Theorem 1.1 and equality case): the rigidity assertion that equality holds if and only if the manifold is CP^n with the Fubini-Study metric requires an explicit argument showing that the Kähler condition plus positive scalar curvature plus systolic equality forces the metric to be the Fubini-Study one. The current sketch appears to invoke uniqueness of the extremal Kähler-Einstein structure without a self-contained reference or derivation ruling out other candidates.
Authors: We agree that the rigidity statement in Theorem 1.1 would be strengthened by a more explicit, self-contained argument. In the revised manuscript, we will expand the equality case analysis to derive directly from the Kähler condition, positive scalar curvature, and systolic equality that the metric must coincide with the Fubini-Study metric on CP^n. This will include the necessary steps establishing uniqueness of the extremal Kähler-Einstein structure in this setting, without relying on external references. revision: yes
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Referee: [§5] §5 (refinements for Fano manifolds): the distinction among the complex quadric, cubic, and quartic via their canonical Kähler-Einstein structures must be shown to follow from the systolic inequalities without circular appeal to the very rigidity statement being proved; a concrete comparison of the relevant 2-systole values or calibration forms is needed.
Authors: We acknowledge the need to avoid any appearance of circularity. In the revision, we will insert explicit comparisons of the 2-systole values (or the associated calibration forms) for the canonical Kähler-Einstein metrics on the quadric, cubic, and quartic. These comparisons will be derived directly from the systolic inequalities, independent of the full rigidity result for CP^n. revision: yes
Circularity Check
No circularity: derivation rests on geometric comparison and calibrated geometry without self-referential reduction
full rationale
The paper derives sharp 2-systole and spherical 2-systole bounds for compact Kähler manifolds with positive scalar curvature, with equality precisely on CP^n equipped with the Fubini-Study metric, plus refinements for certain Fano manifolds. These rest on properties of Kähler metrics, positive scalar curvature, and calibrated geometry rather than any step that defines the target inequality in terms of itself or renames a fitted parameter as a prediction. No load-bearing self-citation chain or uniqueness theorem imported from the same authors is required to close the argument; the equality cases follow from the stated assumptions on the manifold and metric. The derivation is therefore self-contained against external geometric benchmarks.
Axiom & Free-Parameter Ledger
axioms (1)
- domain assumption The manifolds are compact Kähler or spin^c and admit metrics of positive scalar curvature
Lean theorems connected to this paper
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IndisputableMonolith/Foundation/AlexanderDuality.leanalexander_duality_circle_linking unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
Theorem 1.1: max{sys₂(X,ω), sys_π₂(X,ω)} ≤ 4π n(n+1) R̄(ω)⁻¹ with equality iff X ≃ CP^n and ω the Fubini-Study metric
What do these tags mean?
- 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.
- unclear
- Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.
Reference graph
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discussion (0)
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