The Dirichlet problem of homogeneous complex k-Hessian equation in a (k-1)-pesudoconvex domain with isolated singularity

In this paper, we consider the homogeneous complex k-Hessian equation in $\mathrm{\Omega }\mathrm{\backslash }\{0\}$. We prove the existence and uniqueness of the $C^{1,\alpha }$ solution by constructing approximating solutions. The key point for us is to construct the subsolution for approximating problem and establish uniform gradient estimates and complex Hessian estimates which is independent of the approximation.