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

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 $\mathbb{C}\mathrm{P}<sup>n$ and $\mathrm{scal}_g\ge 4n(n+1)$, then $\mathrm{sys}_2^{{\mathrm{st}}(M,g)\le} π$. Moreover, equality holds only for the Fubini-Study metric, up to biholomorphism after choosing the corresponding complex structure. The proof uses Spin$<sup>c$ Dirac operators, a comass estimate for the curvature term in the Lichnerowicz formula, and stable norm-comass duality.

• The paper establishes a sharp stable 2-systolic bound (π) for complex projective spaces under a scalar curvature lower bound, uniquely characterizing the Fubini–Study metric in the equality case.
• It employs Spin^c Dirac operators and calibrated geometry techniques to connect scalar curvature comparisons with stable 2-systolic invariants.
• The work advances scalar curvature rigidity by showing that equality in the systolic bound forces a Kähler–Einstein structure, providing a new geometric rigidity phenomenon.

Sharp Stable 2-Systolic Inequalities, Scalar Curvature, and Spin$^c$ Comass Bounds

Introduction and Context

This work establishes a precise relationship between stable 2-systolic invariants of Riemannian manifolds, lower scalar curvature bounds, and Spin$^c$ geometric analysis. The central achievement is the derivation of a sharp upper bound for the stable 2-systole of complex projective spaces equipped with metrics whose scalar curvature meets or exceeds that of the normalized Fubini–Study metric. The authors connect tools from calibrated geometry, index theory, and curvature estimates, culminating in strong rigidity results: equality in the systolic bound characterizes the Fubini–Study metric, up to biholomorphism.

Main Results

Stable 2-Systoles and Scalar Curvature

The stable 2-systole, $\sys_2^\mathrm{st}(M,g)$, is defined homologically via the stable mass norm on $H_2(M;\mathbb{Z})/\mathrm{torsion}$ as the infimum of normalized areas of nontrivial 2-cycles. It is the dual invariant to the comass norm on $H^2(M;\mathbb{R})$, and, critically, this duality interfaces scalar curvature comparison with geometric measure theory.

The main theorem asserts:

**Let $M$ be diffeomorphic to $\mathbb{CP}^n$ and $g$ a Riemannian metric with $\scal_g \geq 4n(n+1)$. Then

$\sys_2^\mathrm{st}(M,g) \leq \pi$

with equality if and only if $^c$0 is, up to biholomorphism, the normalized Fubini–Study metric.**

This bound is proven sharp, strengthening and extending prior non-sharp estimates for higher-dimensional complex projective spaces, e.g., those of Stryker, and fully generalizing previously known extremal cases in dimension four (via LeBrun’s work).

Rigidity and Equality Cases

If equality occurs, the manifold admits a complex structure and is biholomorphic to the standard complex projective space, with the metric being isometric (in that complex structure) to the Fubini–Study metric. Thus, the Fubini–Study metric is uniquely characterized as the only Riemannian metric on $^c$1 with this scalar curvature lower bound and maximal stable 2-systole.

Odd-Dimensional Analogue

Analogous sharp results are obtained for manifolds diffeomorphic to $^c$2, with scalar curvature bounds corresponding to those of the even-dimensional case. In case of equality, the universal cover is isometric to $^c$3, reflecting the splitting dictated by the existence of a parallel 2-form with 1-dimensional kernel, in accordance with the de Rham theorem.

Techniques and Methods

Spin$^c$4 Dirac Operator and Lichnerowicz Formula

The proof utilizes Spin$^c$5 geometry, particularly the Dirac operator twisted by complex line bundles. The Lichnerowicz formula expresses the square of the Dirac operator in terms containing both the scalar curvature and a Clifford action of the curvature of the twisting bundle. A crucial technical innovation is a sharp Clifford-algebra (comass) bound for this curvature term: the operator norm of the Clifford action of any degree-2 form is bounded from above by the comass times the complex dimension.

Comass-Stable Norm Duality

Exploiting the Federer–Gromov duality between comass (on cohomology) and stable norm (on homology), the inequality for the curvature term becomes a systolic inequality, relating the geometric calibration of 2-forms to area-minimizing cycles in homology.

Specifically, in rank-one scenarios (such as for $^c$6), the systolic bound $^c$7 follows immediately from this duality.

Index-Theoretic Nonvanishing

Topological assumptions are encoded by the calculation of the paired index class: the nontriviality of the topological index for suitable Spin$^c$8 structures is ensured by the non-vanishing Todd genus, reflecting the Fano property of projective space.

Handling Regularity and Rigidity

The equality case, requiring optimizing closed 2-forms in a given cohomology class, is handled by demonstrating regularity up to smoothness and showing that harmonic representatives must be Kähler forms yielding Einstein metrics. The proof employs analytic compactness arguments (with weak convergence in slices of forms and connections) and elliptic regularity.

Notable Numerical and Structural Claims

• Sharpness of $^c$9: For the normalized Fubini–Study metric, each projective line realizes the bound, i.e., its normalized area is precisely $\sys_2^\mathrm{st}(M,g)$0.
• Even-dimensional case: The lower scalar curvature bound is $\sys_2^\mathrm{st}(M,g)$1 with normalization such that each complex line has area $\sys_2^\mathrm{st}(M,g)$2, and the bound is saturated only for the canonical metric.
• Odd-dimensional case: The universal cover splits off a line, enforced by the structure induced from a parallel 2-form with codimension-one kernel; this strictly restricts possible global geometry in the equality case.

Implications and Future Directions

Scalar Curvature Rigidity

These theorems embody a scalar curvature rigidity phenomenon for complex projective spaces (and their products with a circle): the combination of homology, curvature, and calibration uniquely determine the complex and Riemannian structure. This sharpens the scope of rigidity results classically seen for spheres, tori, and Kähler–Einstein Fano manifolds, and specifically harnesses stable systolic invariants as geometric probes.

Systolic Geometry and Generalizations

The techniques presented—based on combining index theory, calibrations, and geometric measure theory—generalize beyond $\sys_2^\mathrm{st}(M,g)$3 or its products and may conjecturally extend to other settings where large symmetry groups or complex structures interact with curvature bounds. It is plausible that similar sharp inequalities could be derived for other homogeneous spaces or certain Kähler–Einstein varieties, provided the topological and cohomological framework supports a strong index-theoretic backbone.

Calibrated Structures and Mass Minimization

The argument confirms that the minimal area 2-cycles (projective lines in $\sys_2^\mathrm{st}(M,g)$4) for the Fubini–Study metric are indeed calibrated, hence mass-minimizing. This connects the systolic extremality to the theory of calibrations, suggesting further directions employing special holonomy or other calibrated geometries.

Spin$\sys_2^\mathrm{st}(M,g)$5 Geometric Estimates

The Clifford-algebraic comass estimates introduced offer new analytic tools for evaluating curvature terms in Spin$\sys_2^\mathrm{st}(M,g)$6 settings. Their optimality and rigidity (forcing the metric to be Kähler–Einstein in the equality case) may have applications in other rigidity problems for scalar curvature and Dirac operators.

Conclusion

This paper provides a complete characterization of the stable 2-systole extremality and rigidity for complex projective spaces under sharp scalar curvature lower bounds. The results decisively identify the Fubini–Study metric as the unique maximizer of the stable 2-systole under this curvature constraint, with deeply intertwined analytic and topological arguments based on Spin$\sys_2^\mathrm{st}(M,g)$7 index theory and comass duality. The methods and results set a new standard for sharp scalar curvature–systole relationships in complex and calibrated geometry, with clear implications for future explorations of rigidity and extremality in Riemannian and complex geometry.

Reference: "Stable $\sys_2^\mathrm{st}(M,g)$8-systoles, scalar curvature and spin$\sys_2^\mathrm{st}(M,g)$9 comass bounds" (2604.25900)