The Surjectivity And Isomorphism Of Dolbeault Cohomologies In Several Complex Variables

In this paper,we adopt the Ohsawa-Takegoshi theory to study the surjectivity problem of Dolbeault cohomology classes in weakly pseudoconvex K?hler manifolds,and the content is taken from[22,23].By means of the existence and Sobolev regularity of （?） equations and extension of （?）-closed forms with Sobolev coefficients,we give the result of isomorphism of Sobolev Dolbeault cohomologies in a domain with concave boundary,and the content is taken from[21].In Chapter 1,we introduce the backgrounds,developments and motivations of the surjectivity and isomorphism problems,respectively.In Chapter 2,we briefly introduce the basic knowledge on Dolbeault cohomology,L² method and （?）-Neumann theory.In Chapter 3,we obtain two L² extension results of （?）-closed forms on weakly pseudoconvex K?hler manifolds,when the holomorphic line bundles involved equipped with smooth metrics satisfying some positivity conditions,and as a direct corollary,the surjectivity of of Dolbeault cohomologies is obtained.In Chapter 4,we give an L² extension result of the Dolbeault cohomology class on weakly pseudoconvex K?hler manifolds,when the holomorphic line bundle involved equipped with a metric with some analytic singularities.In Chapter 5,we give a theorem on the isomorphism of Dolbeault cohomologies with Sobolev coefficients in a bounded smooth domain with concave boundary,and some results on the infinite dimensionality and vanishing of some Sobolev Dolbeault cohomology groups,and the existence and regularity of solutions of （?） equations in the W_*^-1 category by some approximating method.