Ramadanov conjecture and line bundles over compact Hermitian symmetric   spaces
  We compute the Szeg\"o kernels of the unit circle bundles of homogeneous negative line bundles over a compact Hermitian symmetric space. We prove that their logarithmic terms vanish in all cases and, further, that the circle bundles are not diffeomorphic to the unit sphere in $\CC^n$ for Grassmannian manifolds of higher ranks. In particular they provide an infinite family of smoothly bounded strictly pseudo-convex domains on complex manifolds for which the log terms in the Fefferman expansion of the Szeg\"o kernel vanish and which are not diffeomorphic to the sphere. The analogous results for the Bergman kernel are also obtained. 
M. Englis and G.K. Zhang