The rigidity of eigenvalues on Kähler manifolds with positive Ricci lower bound

In this work, optimal rigidity results for eigenvalues on Kähler manifolds with positive Ricci lower bound are established. More precisely, for those Kähler manifolds whose first eigenvalue agrees with the Ricci lower bound, we show that the complex projective space is the only one with the largest multiplicity of the first eigenvalue. Moreover, there is a specific gap between the largest and the second largest multiplicity. In the Kähler--Einstein case, almost rigidity results for eigenvalues are also obtained.