In this article, we prove the transformation formula for reduced Bergman kernels under proper holomorphic correspondences between bounded domains in complex plane. As a corollary, obtain maps. We also establish weighted Finally, provide an application of formula.

We show that the Bergman, Szegő, and Poisson kernels associated to an n-connected domain in the plane are not genuine functions of two complex variables. Rather, they are all given by elementary rational combinations of n+ 1 holomorphic functions of one complex variable and their conjugates. Moreover, all three kernel functions are composed of the same basic n+ 1 functions. Our results can be i...

We construct a pointwise Boutet de Monvel-Sj\"ostrand parametrix for the Szeg\H{o} kernel of weakly pseudoconvex three dimensional CR manifold finite type assuming range its tangential operator to be closed; thereby extending earlier analysis Christ. This particularly extends Fefferman's boundary asymptotics Bergman domains in $\mathbb{C}^{2}$, agreement with D'Angelo's example. Finally our res...

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 Cn for Grassmannian manifolds of higher ranks. In particular they provide an infinite family of smoothly bounded strictly ...

We provide a simple proof of result Rouby-Sj\"ostrand-Ngoc \cite{RSN} and Deleporte \cite{Deleporte}, which asserts that if the K\"ahler potential is real analytic then Bergman kernel an \textit{analytic kernel} meaning its amplitude symbol} phase given by polarization potential. This in particular shows case accepts asymptotic expansion fixed neighborhood diagonal with exponentially small rema...