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Plurisubharmonicity of envelopes of disc functionals on manifolds

2003; De Gruyter; Volume: 2003; Issue: 555 Linguagem: Inglês

10.1515/crll.2003.013

1435-5345

Finnur Lárusson, Ragnar Sigurðsson,

Advanced Algebra and Geometry

We show that a disc functional on a complex manifold has a plurisubharmonic envelope if all its pullbacks by holomorphic submersions from domains of holomorphy in affine space do and it is locally bounded above and upper semicontinuous in a certain weak sense.For naturally defined classes of disc functionals on manifolds, this result reduces a property somewhat stronger than having a plurisubharmonic envelope to the affine case.The proof uses a recent Stein neighbourhood construction of Rosay, who proved the plurisubharmonicity of the Poisson envelope on all manifolds.As a consequence, the Riesz envelope and the Lelong envelope are plurisubharmonic on all manifolds; for the former, we make use of new work of Edigarian.The basic theory of the three main classes of disc functionals is thereby extended to all manifolds.