正 Ricci 下界的 Kähler 流形上的谱几乎刚性
Spectral almost rigidity on Kähler manifolds with positive Ricci lower bound
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中文总结 AI 辅助
本文证明了紧 Kähler 流形上谱几乎刚性的尖锐定理，通过前 n^2+3 个特征值夹逼确定双全纯类型，并给出临界特征值重数的上界与达到条件。
中文摘要 AI 辅助
本文建立了尖锐的 Kähler 谱几乎刚性定理，解决了 Chu--Wang--Zhang 的猜想 1.8。对于满足 $\Ric(\omega)\geq\omega$ 的紧 Kähler 流形，将前 $n^2+3$ 个非零复特征值夹逼到 1 会迫使流形在 Gromov-Hausdorff 意义下接近归一化的复射影空间，并确定了双全纯类型。较小的指标 $n^2+1$ 是非塌缩的尖锐阈值。更一般地，若归一化测度极限具有本质实维数 $r$，则临界特征值的重数至多为 $r+\lfloor r/2\rfloor^2$，且在每一维数中均可达到等号。对复梯度 Gram 矩阵的核投影的一致能量估计导致了整个谱分解上的李代数作用。
英文摘要
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 $\Ric(ω)\geqω$, 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 $r+\lfloor r/2\rfloor^2$, 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.
