Ramadanov conjecture and line bundles over compact Hermitian symmetric spaces

We compute the Szegö 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 ${mathbb C^n}We compute the Szeg? 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 mathbb Cⁿ{mathbb C^n} for Grassmannian manifolds of higher ranks. In particular, they provide an infinite family of smoothly bounded strictly pseudoconvex domains on complex manifolds for which the logarithmic term in the Fefferman expansion of the Szeg? kernel vanishes but whose boundary is not diffeomorphic to the sphere (in fact, it is not even locally spherical). The analogous results for the Bergman kernel are also obtained.