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 $\mathbb{C}{\mathrm{P}}^n$ 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.

20 pages