Spectral almost rigidity on Kähler manifolds with positive Ricci lower bound

In this work, a sharp Kähler spectral almost-rigidity theorem was established, resolving Conjecture 1.8 of Chu–Wang–Zhang. For compact Kähler manifolds satisfying $\text{Ric}(\omega )\ge \omega$, pinching the first $n^2+3$ nonzero complex eigenvalues to one forces the manifolds to be Gromov Hausdorff close to normalized complex projective space and determines the biholomorphism type. The smaller index $n^2+1$ is the sharp threshold for noncollapsing. More generally, if a normalized measured limit has essential real dimension $r$, then the multiplicity of the critical eigenvalue is at most $r+\lfloor r\mathrm{/}2{\rfloor }^2$, with equality attained in every dimension. A uniform energy estimate for kernel projections of complex gradient Gram matrices leads to a Lie algebra action on the whole spectral resolution.