<i>L^P</i> Curvature and the Cauchy-Riemann equation near an isolated singular point

Abstract

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