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We show the vanishing of the log-term in the Fefferman expansion of the Bergman kernel of the disk bundle over a compact simply-connected homogeneous Kähler-Einstein manifold of classical type. Our results extends that in (Engliš and Zhang, Math Z 264(4):901-912, 2010) for the case of Hermitian symmetric spaces of compact type. [ABSTRACT FROM AUTHOR]

Let ( M, g) be a real analytic Kähler manifold. We say that a smooth map Exp : W → M from a neighbourhood W of the origin of T M into M is a diastatic exponential at p if it satisfies where D is Calabi's diastasis function at p (the usual exponential exp obviously satisfied these equations when D is replaced by the square of the geodesics distance from p). In this paper we prove that for every point p of an Hermitian symmetric space of noncompact type M there exists a globally defined diastatic exponential centered in p which is a diffeomorphism and it is uniquely determined by its restriction to polydisks. An analogous result holds true in an open dense neighbourhood of every point of M, the compact dual of M. We also provide a geometric interpretation of the symplectic duality map (recently introduced in Di Scala and Loi (Adv Math 217:2336-2352, 2008)) in terms of diastatic exponentials. As a byproduct of our analysis we show that the symplectic duality map pulls back the reproducing kernel of M to the reproducing kernel of M. [ABSTRACT FROM AUTHOR]