37 Pages

1Potential Theory in Several Complex Variables



Niveau: Supérieur, Doctorat, Bac+8
1Potential Theory in Several Complex Variables Jean-Pierre Demailly Universite de Grenoble I, Institut Fourier, BP 74 URA 188 du C.N.R.S., 38402 Saint-Martin d'Heres, France This work is the second part of our survey article on Monge-Ampere operators. We are concerned here with the theory of complex n-dimensional capacities generalizing the usual logarithmic capacity in C, and with the related notions of pluripolar and negligible sets. Decisive progress in the theory have been made by Bedford-Taylor [B-T1], [B-T2]. The present exposition, which is an expansion of lectures given in Nice at the Centre International de Mathematiques Pures et Appliquees (ICPAM) in 1989, borrows much to these papers. The last section on comparison of capacities is based on the work of Alexander and Taylor [A-T]. We are indebted to Z. B locki and D. Coman for pointing out a few mistakes in the original version. Z. B locki also suggested to derive the improved logarithmic growth estimate (due independently to H. El Mir and J. Siciak) from our proof of Josefson's theorem on the equivalence between locally pluripolar and globally pluripolar sets. 10. Capacities, Regularity and Capacitability The goal of this section is to discuss a few fundamental notions and results of capacity theory.

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