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 CPn up to holomorphic isometry among compact connected Kähler manifolds, for every fixed pair of integers (p,n) with 0≤ p≤ 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.
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BibTeX
@misc{liu2026proof,
title = {{A proof of the spectral rigidity for complex projective space}},
author = {Kefeng Liu and Hao Xu},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03536},
eprint = {2611.03536},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03536},
}
RIS
TY - UNPB
TI - A proof of the spectral rigidity for complex projective space
AU - Kefeng Liu
AU - Hao Xu
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03536
UR - https://arxiv.org/abs/2611.03536
AB - 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≤ p≤ 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.
ER -