An Analytic Proof Of The Geometric Quantization Conjecture Of Guillenmin-Sternberg

Geometric quantizations and symplectic reductions are important topics in symplectic geometry.Firstly,this paper introduces some concepts with respect to symplectic man-ifolds,for instance almost complex manifolds,symplectic manifolds,Dirac operator and Hamiltonian actions;then we define a deformation of Dirac operators.Secondly,a localized estimate of?^-1?0?is given.Finally,we provide a detailed proof of the main theorems.This paper is divided into four chapters:Chapter I gives the background and outlines our research result for the related conclusions concerning geometric quantizations and symplectic reductions;Chapter II gives the preliminary konwledge;Chapter III gives provides our detailed proof of the main theorem.This analytic technique is used to give a a localized estimate of?^-1?0?.Chapter?provides a detailed proof of geometric quantizations and symbiotic transformation,what's more,this method is applied to process a generalization of geometric quantizations and symplectic reductions.