Let \(\Omega \subset \mathbb{C}^n\) be a smoothly bounded strictly pseudoconvex domain. The boundary \(\partial\Omega\) is said to be obstruction flat if the log singularity (the obstruction function) in the logarithmic potential of the complete K\"ahler--Einstein metric on \(\Omega\) vanishes. It is called Bergman logarithmically flat if the logarithmic singularity in the Fefferman expansion of the Bergman kernel vanishes. Both notions of flatness depend only on the local CR geometry of the boundary and can be defined for any strictly pseudoconvex real hypersurface.

In this talk, we consider real hypersurfaces arising as unit circle bundles of negative Hermitian line bundles over a complex manifold \(M\). We study the relationship between obstruction flatness and Bergman logarithmic flatness of these circle bundles and the K\"ahler geometry of the induced metric on \(M\). The talk is based on joint work with Peter Ebenfelt and Hang Xu.