Kähler Duality and Projective Embeddings

Motivated by the duality theory between Hermitian symmetric spaces of noncompact and compact types, we introduce and examine the concept of Kähler duality between exact domains of C n (cf.Definitions 2 and 3).Specifically, we address the following question (cf.Definition 1).Question.Let (U, g) be an exact domain admitting a Kähler dual (U * , g * ).Assume that there exists α > 0 such that the following conditions hold:(a) (U, αg) has a Fubini-Study completion;(b) (U * , αg * ) has a Fubini-Study compactification.Is it true that (U, g) is biholomorphically isometric to a bounded symmetric domain?In Theorem 1.1, we affirmatively answer this question for a Cartan-Hartogs domain M Ω,µ equipped with an appropriate metric g Ω,µ .In Theorem 1.2, we provide a counterexample by introducing another Kähler metric ĝΩ,µ on M Ω,µ , that satisfies both conditions (a) and (b).Furthermore, in Theorem 1.3, we characterize Hermitian symmetric spaces of compact type among classical flag manifolds based solely on Kähler duality.We also propose and thoroughly investigate a conjecture that our question has a positive answer if the metric g is Kähler-Einstein.