Ramadanov conjecture and line bundles over compact Hermitian symmetric spaces

TL;DR

This paper computes Szeg"o kernels for certain line bundles over compact Hermitian symmetric spaces, showing their log terms vanish and identifying topological differences from spheres, thus providing new examples of pseudo-convex domains.

Contribution

It demonstrates the vanishing of log terms in Szeg"o kernels for a class of line bundles and shows these domains are not diffeomorphic to spheres, expanding understanding of complex geometric structures.

Findings

Logarithmic terms in Szeg"o kernels vanish for these domains.

Circle bundles are not diffeomorphic to spheres in higher-rank Grassmannians.

Provides infinite families of pseudo-convex domains with vanishing log terms.

Abstract

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.