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An improved cowaist inequality for line bundles and consequences
Pith reviewed 2026-05-07 11:42 UTC · model grok-4.3
The pith
A refinement of the cowaist inequality for Hermitian line bundles yields sharp stable two-systolic inequalities for odd-dimensional complex projective spaces.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The paper proves a refinement of the cowaist inequality in the case of Hermitian line bundles. Combined with the cowaist-systole estimates in recent work, this gives sharp stable two-systolic inequalities for odd-dimensional complex projective spaces. For products of n two-dimensional spheres with scalar curvature at least 2n, it improves the upper bound on the stable two systole from O(n^4 log n) to O(n^3 log n). In particular, for the product of two two-spheres with scalar curvature at least 4, it improves the stable two systole upper bound from 36 pi to 12 pi.
What carries the argument
The refined cowaist inequality for Hermitian line bundles, which tightens the relation between cowaist and systole quantities.
Load-bearing premise
The cowaist refinement applies specifically to Hermitian line bundles and combines directly with prior cowaist-systole estimates without further restrictions on the manifolds or bundles.
What would settle it
An explicit computation of the stable two-systole for three-dimensional complex projective space that falls below the predicted sharp value would falsify the combined inequality.
read the original abstract
We prove a refinement of Gromov's cowaist inequality in the case of Hermitian line bundles. Combined with the cowaist-systole estimates in recent work of Stryker, this gives sharp stable two-systolic inequalities for odd-dimensional complex projective spaces. For products of $n$ two-dimensional spheres with scalar curvature at least $2n$, it improves the upper bound on the stable two systole from $O(n^4\log n)$ to $O(n^3\log n)$. In particular, for $S^2\times S^2$ with scalar curvature at least $4$, it improves the stable two systole upper bound from $36\pi$ to $12\pi$.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript proves a refinement of Gromov's cowaist inequality adapted to Hermitian line bundles on Riemannian manifolds. When combined with the cowaist-systole estimates from Stryker's recent work, this yields sharp stable two-systolic inequalities for odd-dimensional complex projective spaces. For products of n two-spheres with scalar curvature bounded below by 2n, the refinement improves the upper bound on the stable two-systole from O(n^4 log n) to O(n^3 log n); in particular, for S^2 × S^2 with scalar curvature at least 4 the bound improves from 36π to 12π.
Significance. If the refined inequality holds, the work strengthens systolic geometry by delivering sharp stable two-systolic bounds on CP^{2k+1} and improved quantitative estimates on sphere products. The adaptation of Gromov's argument to the Hermitian setting and the direct plug-in to Stryker's estimates constitute a clean advance that tightens existing constants without additional curvature or topological hypotheses.
minor comments (3)
- §2, after the statement of the refined cowaist inequality: the notation for the Hermitian metric h and the associated curvature form should be introduced explicitly before the volume-waist comparison is stated, to avoid ambiguity when the reader compares the new bound with Gromov's original formulation.
- §4, Theorem 4.2: the dependence of the constant in the O(n^3 log n) bound on the lower scalar-curvature threshold 2n is not made fully explicit; a short remark clarifying whether the implied constant is independent of n would strengthen the statement.
- References: the citation to Stryker's cowaist-systole paper should include the arXiv number or journal details for immediate accessibility.
Simulated Author's Rebuttal
We thank the referee for the positive assessment of our manuscript, the clear summary of its contributions, and the recommendation for minor revision. No specific major comments were raised in the report.
Circularity Check
No significant circularity in the derivation chain
full rationale
The paper refines Gromov's cowaist inequality by adapting the original argument to Hermitian line bundles, producing a stricter volume-waist relation that holds without additional curvature assumptions. This refined bound is then plugged directly into Stryker's independent cowaist-systole estimates to derive new systolic inequalities for odd-dimensional CP^n and sphere products. No load-bearing step reduces by construction to a self-definition, a fitted parameter renamed as a prediction, or a self-citation chain; all central claims rest on external prior results or explicit adaptation of Gromov's method, making the derivation self-contained against external benchmarks.
Axiom & Free-Parameter Ledger
axioms (2)
- domain assumption Gromov's cowaist inequality holds for general manifolds
- domain assumption Stryker's cowaist-systole estimates are valid and applicable
Reference graph
Works this paper leans on
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[1]
Stable $2$-systoles, scalar curvature and spin$^c$ comass bounds
arXiv:2604.25900. [GHK23] T. Goodwillie, J. Hebda, and M. Katz. Extending Gromov’s optimal systolic inequality.J. Geom., 114:Paper No. 23,
work page internal anchor Pith review Pith/arXiv arXiv
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[2]
[Gro23] M. Gromov. Four lectures on scalar curvature. InPerspectives in scalar curvature. Vol. 1, pages 1–514. World Sci. Publ., Hackensack, NJ, [2023]©2023. [Str26] D. Stryker. Stable 2-systole bounds in positive scalar curvature,
2023
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[3]
Stable 2-systole bounds in positive scalar curvature
arXiv:2604.22106. Department of Mathematics, University of Maryland, 4176 Campus Dr, College Park, MD 20742, USA
work page internal anchor Pith review Pith/arXiv arXiv
discussion (0)
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