The spin $^{\bf c}$ Dirac operator on high tensor powers of a line bundle

Abstract

We study the asymptotic of the spectrum of the ${\rm spin^c}$ Dirac operator on high tensor powers of a line bundle. As application, we get a simple proof of the main result of Guillemin–Uribe [13, Theorem 2], which was originally proved by using the analysis of Toeplitz operators of Boutet de Monvel and Guillemin [10].