Asymptotics of small eigenvalues on degenerations of K\"ahler manifolds

Abstract

We derive the exact asymptotic rates of the small eigenvalues of the Laplacian on one-parameter degenerations of compact K\"ahler 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-Amp\`ere equations, introduced by Guo--Phong--Song--Sturm--Tong and adapted by Guedj--T\^o. As an application, we establish estimates for degenerations of compact K\"ahler manifolds with reducible singular fibers.