{$L\sp p$}-curvature and the Cauchy-Riemann equation near an isolated singular point

Abstract

Let $X$ be a complex $n$-dimensional reduced analytic space with isolated singular point $x_{0}$, and with a strongly plurisubharmonic function $\rho:X \to [0,\infty)$ such that $\rho(x_{0}) = 0$. A smooth Kähler form on $X\setminus\{x_{0}\}$ is then defined by ${\bf i}\partial\bar{\partial}\rho$. The associated metric is assumed to have $L^{n}_{\rm{loc}}$-curvature, to admit the Sobolev inequality and to have suitable volume growth near $x_{0}$. Let $E \to X\setminus\{x_{0}\}$ be a Hermitian-holomorphic vector bundle, and $\xi$ a smooth $(0,1)$-form with coefficients in $E$. The main result of this article states that if $\xi$ and the curvature of $E$ are both $L^{n}_{\rm{loc}}$, then the equation $\bar{\partial}u = \xi$ has a smooth solution on a punctured neighbourhood of $x_{0}$. 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.