Asymptotics of small eigenvalues on degenerations of Kähler manifolds

We derive the exact asymptotic rates of the small eigenvalues of the Laplacian on one-parameter degenerations of compact Kähler manifolds equipped with induced background metrics. This generalizes a recent result of Dai and Yoshikawa to higher dimensions. To achieve this, we combine Li’s uniform Skoda inequality with the method of auxiliary Monge-Ampère equations, introduced by Guo–Phong–Song–Sturm–Tong and adapted by Guedj–Tô. As an application, we establish estimates for degenerations of compact Kähler manifolds with reducible singular fibers.