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Lp-curvature and the Cauchy-Riemann equation near an isolated singular point

Published online by Cambridge University Press:  22 January 2016

Adam Harris  and
Yoshihiro Tonegawa
Adam Harris
Affiliation:
School of Mathematics and Statistics, University of Melbourne, Parkville, VIC 3052, Australia, A.Harris@ms.unimelb.edu.au
Yoshihiro Tonegawa
Affiliation:
Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan, tonegawa@@math.sci.hokudai.ac.jp
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Abstract

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Let X be a complex n-dimensional reduced analytic space with isolated singular point x0, and with a strongly plurisubharmonic function ρ : X → [0, ∞) such that ρ(x0) = 0. A smooth Kähler form on X \ {x0} is then defined by i∂∂ρ. The associated metric is assumed to have -curvature, to admit the Sobolev inequality and to have suitable volume growth near x0. Let E → X \ {x0} be a Hermitian-holomorphic vector bundle, and ξ a smooth (0, 1)-form with coefficients in E. The main result of this article states that if ξ and the curvature of E are both then the equation has a smooth solution on a punctured neighbourhood of x0. Applications of this theorem to problems of holomorphic extension, and in particular a result of Kohn-Rossi type for sections over a CR-hypersurface, are discussed in the final section.

Keywords

32D2032L10
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 164 , December 2001 , pp. 35 - 51
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2001

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