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Abstract. We study the complete Kähler-Einstein metric of a Hartogs domain Ω built on an irreducible bounded symmetric domain Ω, using a power N µ of the generic norm of Ω. The generating function of the Kähler-Einstein metric satisfies a complex Monge-Ampère equation with boundary condition. The domain Ω is in general not homogeneous, but it has a subgroup of automorphisms, the orbits of which are parameterized by X ∈ [0, 1[. This allows to reduce the Monge-Ampère equation to an ordinary differential equation with limit condition. This equation can be explicitly solved for a special value µ 0 of µ. We work out the details for the two exceptional symmetric domains. The special value µ 0 seems also to be significant for the properties of other invariant metrics like the Bergman metric; a conjecture is stated, which is proved for the exceptional domains.

We determine the group of diffeomorphisms of a bounded symmetric domain, which preserve simultaneously the hyperbolic and the flat symplectic form.

Abstract. We introduce the notion of virtual Bergman kernel and study some of its applications. ContentsIntroduction 1 1. Virtual Bergman kernels 2 2. Virtual Bergman kernels for bounded symmetric domains 7 3. Tables for bounded symmetric domains 9 4. Open problems 11 References 12 IntroductionLet Ω ⊂ C n be a domain and p : Ω →]0, +∞[ a weight function on Ω. Consider the "inflated domains"where is the standard Hermitian norm on C m . In our joint work [1] with Yin Weiping, we computed explicitly the Bergman kernel of some "egg domains"; among them Ω 1 , when Ω is a bounded symmetric domain and p a real power of the generic norm of Ω. We then obtained the Bergman kernel of the corresponding Ω m by using the "inflation principle" of [2], which allows to deduce (for any weight function p) the Bergman kernel of Ω m from the Bergman kernel of Ω 1 . The "inflation principle" says that if the Bergman kernel of Ω 1 is

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