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Mesures de Monge-Ampère et mesures pluriharmoniques. (Monge-Ampère measures and pluriharmonic measures). (French) Zbl 0595.32006
Let \(\Omega\) be a relatively compact open subset in a Stein manifold, and \(n=\dim_{{\mathbb{C}}}\Omega\). Assume that \(\Omega\) is hyperconvex, i.e. that there exists a bounded psh (plurisubharmonic) exhaustion function on \(\Omega\). A ”pluricomplex Green function” \(u_{\Omega}\) is then naturally defined on \(\Omega \times \Omega:\) For all \(z\in \Omega\), \(u_ z(\zeta):= u_{\Omega}(z,\zeta)\) is the solution of the Dirichlet problem for the complex Monge-Ampère equation \((dd^ cu_ z)^ n=0\) on \(\Omega\) \(\setminus \{z\}\) such that \(u_ z(\zeta)=\log | \zeta - z| +O(1)\) at \(\zeta =z\); \(u_{\Omega}\) is shown to be continuous outside the diagonal and invariant under biholomorphisms. Bedford and Taylor’s Monge-Ampère operators are used in conjunction with a general Lelong-Jensen formula previously found by the author [Mem. Soc. Math. Fr., Nouv. Ser. 19, 124 p. (1985; Zbl 0579.32012)] in order to construct an invariant pluricomplex Poisson kernel \(d\mu_ z(\zeta):= (2\pi)^{- n}(dd^ cu_ z(\zeta))^{n-1}\wedge d^ cu_ z(\zeta)|_{\partial \Omega},\) \((z,\zeta)\in \Omega \times \partial \Omega\). Each measure \(\mu_ z\) on \(\partial \Omega\) is such that \(\mu_ z(V)=V(z)\) for every function V pluriharmonic on \(\Omega\) and continuous on \({\bar \Omega}\); furthermore, \(\mu_ z\) is carried by the set of strictly pseudoconvex points of \(\partial \Omega\) if \({\bar \Omega}\) has a \(C^ 2\) psh defining function. The principal part of the singularity of \(d\mu_ z(\zeta)\) on the diagonal of \(\partial \Omega\) is then computed explicitly when \(\Omega\) is strictly pseudoconvex, using an osculation of \(\partial \Omega\) by balls. Through a complexification process, it is finally shown that Monge-Ampère measures provide an explicit formula representing every point of a convex compact subset \(K\subset {\mathbb{R}}^ n\) as a barycenter of the extremal points of K.

MSC:
32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)
32C30 Integration on analytic sets and spaces, currents
32F45 Invariant metrics and pseudodistances in several complex variables
32U05 Plurisubharmonic functions and generalizations
31C10 Pluriharmonic and plurisubharmonic functions
32E10 Stein spaces
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References:
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