The Green Function With Several Poles Of Complex Hessian Operator

The research of Pluripotential theory depends on the study of complex Monge-Amp?re operator.In recent years,there are a lot of results about complex Monge-Amp?re operator,and they have many applications used in complex analysis and complex differential geometry.Corresponding with complex Monge-Amp?re operator,many experts and scholars have obtained a lot of results on the potential theory of real Hessian operators.In this paper,we apply the methods in the potential theory of complex Monge-Amp?re operator and real Hessian operator to study complex Hessian operator.We study the Green function of single pole and several poles of complex Hessian equation,and prove the continuity of Green function.We proved that the Green function is the unique solution of the Dirichlet problem of complex Hessian equations on m-hyperconvex domains in complex space~n.