A proof of the spectral rigidity for complex projective space

Abstract

We give a proof of a folklore conjecture that a compact Kähler manifold with the same scalar Laplace spectrum as complex projective space equipped with the Fubini--Study metric is holomorphically isometric to it. More generally, we prove that the spectrum of the Hodge Laplacian in any one fixed degree $p$ determines $\CP^n$ up to holomorphic isometry among compact connected Kähler manifolds, for every fixed pair of integers $(p,n)$ with $0\leq p\leq n$. Equality of a single Hodge spectrum first forces periodic geodesic flow and the integral cohomology ring of complex projective space. The ordinary first heat coefficient then identifies the total scalar curvature except on a Pell sequence of pairs. For those pairs, a clean wave-trace argument, a Krawtchouk estimate, and a mod-two projective-bundle obstruction determine the geodesic return holonomy. A canonical Kähler involution recovers a weighted first heat coefficient which is nonzero throughout the exceptional sequence. The Kobayashi--Ochiai theorem and a fixed-class second-heat-coefficient calculation complete the argument.