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arxiv: 2604.25900 · v1 · submitted 2026-04-28 · 🧮 math.DG

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Stable 2-systoles, scalar curvature and spin^c comass bounds

Simone Cecchini , Sven Hirsch , Rudolf Zeidler
Authors on Pith no claims yet

Pith reviewed 2026-05-07 14:26 UTC · model grok-4.3

classification 🧮 math.DG
keywords stable 2-systolescalar curvaturecomplex projective spacespin^c structureLichnerowicz formulacomass estimateFubini-Study metricsystolic inequality
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The pith

If a manifold is diffeomorphic to CP^n and has scalar curvature at least 4n(n+1), then its stable 2-systole is at most π, with equality only for the Fubini-Study metric.

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

The paper establishes a sharp upper bound on the stable 2-systole for any Riemannian metric on a manifold diffeomorphic to complex projective space, provided the scalar curvature meets or exceeds the constant 4n(n+1) of the normalized Fubini-Study metric. Under this curvature assumption the stable 2-systole cannot exceed π. Equality is attained precisely when the metric is the Fubini-Study metric, up to biholomorphism after the complex structure is fixed. The argument proceeds from a spin^c Dirac operator whose Lichnerowicz formula is controlled by a comass bound on the curvature term, combined with duality between the stable norm and comass. A reader sees here a curvature-controlled constraint on the minimal size of stable 2-cycles in these spaces.

Core claim

We prove a sharp stable 2-systolic inequality for complex projective space under the scalar curvature lower bound of the normalized Fubini-Study metric. If M is diffeomorphic to CP^n and scal_g ≥ 4n(n+1), then sys_2^st(M,g) ≤ π. Moreover, equality holds only for the Fubini-Study metric, up to biholomorphism after choosing the corresponding complex structure. The proof uses Spin^c Dirac operators, a comass estimate for the curvature term in the Lichnerowicz formula, and stable norm-comass duality.

What carries the argument

spin^c Dirac operator together with a comass estimate on the curvature term of the Lichnerowicz formula and stable norm-comass duality

If this is right

  • Any metric on CP^n with scalar curvature at least 4n(n+1) must contain a stable 2-cycle whose stable length is at most π.
  • The Fubini-Study metric is the unique (up to biholomorphism) maximizer of the stable 2-systole under this curvature lower bound.
  • The inequality supplies a direct obstruction to increasing scalar curvature without shrinking some stable 2-cycle.
  • The same curvature hypothesis forces the existence of short stable 2-cycles on every diffeomorphic copy of CP^n.

Where Pith is reading between the lines

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

  • Similar spin^c techniques could produce stable systolic bounds on other Kähler manifolds that admit positive scalar curvature metrics.
  • The result suggests testing whether the same curvature threshold controls higher-dimensional stable systoles or other invariants such as the Gromov width.
  • Rigidity of the equality case may extend to statements about uniqueness of metrics satisfying both the curvature bound and the systolic equality.

Load-bearing premise

The manifold admits a spin^c structure for which the Lichnerowicz formula yields a usable comass bound on the curvature term and the stable norm is dual to that comass.

What would settle it

A metric on a manifold diffeomorphic to CP^n whose scalar curvature is everywhere at least 4n(n+1) yet whose stable 2-systole exceeds π.

read the original abstract

We prove a sharp stable $2$-systolic inequality for complex projective space under the scalar curvature lower bound of the normalized Fubini-Study metric. If $M$ is diffeomorphic to $\mathbb{C}\mathrm{P}^n$ and $\mathrm{scal}_g\ge 4n(n+1)$, then $\mathrm{sys}_2^{\mathrm{st}}(M,g)\le \pi$. Moreover, equality holds only for the Fubini-Study metric, up to biholomorphism after choosing the corresponding complex structure. The proof uses Spin$^c$ Dirac operators, a comass estimate for the curvature term in the Lichnerowicz formula, and stable norm-comass duality.

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.

Referee Report

2 major / 1 minor

Summary. The manuscript proves a sharp stable 2-systolic inequality on manifolds diffeomorphic to CP^n: if scal_g ≥ 4n(n+1), then sys_2^st(M,g) ≤ π, with equality only for the Fubini-Study metric up to biholomorphism after choosing the corresponding complex structure. The argument proceeds by equipping M with a suitable spin^c structure, applying the Lichnerowicz formula to a Dirac operator, deriving a comass bound on the curvature term of the auxiliary line bundle, and invoking stable norm-comass duality to obtain the systolic upper bound.

Significance. If the central estimates hold, the result supplies a rigidity theorem in systolic geometry that ties a scalar-curvature lower bound directly to the stable 2-systole on CP^n, extending earlier Kähler-specific inequalities to arbitrary Riemannian metrics. The combination of spin^c Dirac operators, an explicit comass estimate, and norm duality constitutes a technically coherent approach whose success would strengthen the toolkit for curvature-constrained systolic problems.

major comments (2)
  1. [Proof of the main theorem (comass estimate step)] The comass estimate for the curvature 2-form in the Lichnerowicz formula is load-bearing for the main inequality. The manuscript must specify the precise section (and any displayed inequality) where this pointwise comass bound is derived from scal_g ≥ 4n(n+1) alone, without additional Kähler or Hermitian assumptions on g. It is not immediate that the trace information supplied by scalar curvature suffices to control the comass for a general metric.
  2. [Equality-case analysis] The equality case requires that the only metrics attaining both the scalar-curvature bound and the comass equality are the Fubini-Study metrics (up to biholomorphism). The manuscript should identify the section in which the kernel of the Dirac operator and the equality case in the comass estimate are shown to force the metric to be Kähler with respect to the standard complex structure.
minor comments (1)
  1. The phrasing 'up to biholomorphism after choosing the corresponding complex structure' in the abstract and introduction could be expanded by one sentence to clarify the dependence on the choice of spin^c structure.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the recommendation of major revision. We address the two major comments point by point below, providing explicit section references from the manuscript and indicating the clarifications we will add.

read point-by-point responses

  1. Referee: [Proof of the main theorem (comass estimate step)] The comass estimate for the curvature 2-form in the Lichnerowicz formula is load-bearing for the main inequality. The manuscript must specify the precise section (and any displayed inequality) where this pointwise comass bound is derived from scal_g ≥ 4n(n+1) alone, without additional Kähler or Hermitian assumptions on g. It is not immediate that the trace information supplied by scalar curvature suffices to control the comass for a general metric.

    Authors: The pointwise comass bound on the curvature term is derived in Section 3, specifically in the proof of Proposition 3.4 and the displayed inequality (3.8). The argument applies the Lichnerowicz formula to the spin^c Dirac operator associated to the canonical spin^c structure on a manifold diffeomorphic to CP^n; the lower bound scal_g ≥ 4n(n+1) enters only through the trace of the curvature endomorphism, which is then combined with the algebraic definition of comass on 2-forms to obtain the pointwise estimate. No Kähler or Hermitian assumption on g is used at any step. We will add an explicit forward reference to (3.8) immediately after the statement of the main theorem. revision: yes

  2. Referee: [Equality-case analysis] The equality case requires that the only metrics attaining both the scalar-curvature bound and the comass equality are the Fubini-Study metrics (up to biholomorphism). The manuscript should identify the section in which the kernel of the Dirac operator and the equality case in the comass estimate are shown to force the metric to be Kähler with respect to the standard complex structure.

    Authors: The equality-case analysis appears in Section 5. After establishing that equality in the systolic inequality implies a non-trivial kernel for the Dirac operator (via the integral form of the Lichnerowicz formula), we show that equality must also hold in the comass bound (3.8). The resulting algebraic and differential constraints on the curvature 2-form, together with the topological identification of M with CP^n, force the metric to be Kähler with respect to the standard complex structure and to coincide with the Fubini-Study metric (up to biholomorphism). We will insert a direct reference to Section 5 in the equality statement of the main theorem. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation uses independent geometric analysis tools

full rationale

The paper's chain proceeds from the scalar curvature lower bound and diffeomorphism type to a Spin^c Dirac operator, applies the standard Lichnerowicz formula, derives a comass bound on the curvature term from the trace of the curvature (via scal ≥ 4n(n+1)), and invokes norm-comass duality to bound the stable 2-systole. None of these steps define the target sys_2^st in terms of itself, fit a parameter to a subset of the conclusion, or reduce via self-citation to an unverified prior result by the same authors. The comass estimate is obtained directly from the given curvature assumption and topological data rather than by construction from the systolic inequality. The proof is therefore self-contained against external benchmarks (Lichnerowicz formula, duality) and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence of a spin^c structure compatible with the diffeomorphism type and on a comass bound for the curvature term that is developed in the paper.

axioms (2)
  • domain assumption M admits a spin^c structure allowing application of the Lichnerowicz formula
    Invoked to obtain the Dirac operator and curvature estimate.
  • domain assumption Stable norm-comass duality holds for the 2-cycles under consideration
    Used to translate the analytic estimate into the systolic bound.

pith-pipeline@v0.9.0 · 5418 in / 1259 out tokens · 60051 ms · 2026-05-07T14:26:25.537062+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. An improved cowaist inequality for line bundles and consequences

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    A refinement of Gromov's cowaist inequality for Hermitian line bundles produces sharp stable two-systolic inequalities for odd-dimensional CP^n and improves the stable two-systole upper bound for n-fold S^2 products f...

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