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

TL;DR

This paper establishes a sharp inequality relating scalar curvature and stable 2-systoles on complex projective spaces, characterizing the Fubini-Study metric as the unique extremizer.

Contribution

It proves a new sharp stable 2-systolic inequality for complex projective spaces under scalar curvature bounds, using Spin^c Dirac operators and curvature estimates.

Findings

The inequality is sharp and attained only by the Fubini-Study metric.

Equality characterizes the Fubini-Study metric up to biholomorphism.

The proof combines Spin^c techniques with comass and stable norm duality.

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.