  In this work, optimal rigidity results for eigenvalues on K\"ahler manifolds
with positive Ricci lower bound are established. More precisely, for those
K\"ahler 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 K\"ahler--Einstein
case, almost rigidity results for eigenvalues are also obtained.
