Spectral properties of Kähler quotients

Abstract

The asymptotic behavior for the spectral measure of a Kähler manifold has been studied by many authors in the context of Kähler quantization. It is well known that the spectral measure has an asymptotic expansion, while the coefficients of this expansion are not known even for very simple examples. In this thesis we study the spectral properties of Kähler manifolds assuming the existence of some symmetry, i.e., a Hamiltonian action. The main tool we will use is a function which we call the stability function. Roughly speaking, it is the function which compares quantum states before reduction with quantum states after reduction. We will study this function in detail, compute the function for many classes of Kähler manifolds, and apply it to study various spectral problems on Kähler quotients. As for the spectral measure, we will give an explicit way to compute the coefficients in the asymptotic expansion for toric varieties. It turns out that the upstairs spectral measure in this case is described by an interesting integral transform which we will call the twisted Mellin transform. We will study both analytic and combinatorial aspects of this transform in the beginning of this thesis.