Problems on the geometric function theory in several complex variables and complex geometry - Yuan, Yuan (ProQuest Dissertations & Theses, 2010)
The thesis consists of two parts. In the first part, we study the rigidity for the local holomorphic isometric embeddings. On the one hand, we prove the total geodesy for the local holomorphic conformal embedding from the unit ball of complex dimension at least 2 to the product of unit balls and hence the rigidity for the local holomorphic isometry is the natural corollary. Before obtaining the total geodesy, the algebraic extension theorem is derived following the idea in [MN] by considering the sphere bundle of the source and target domains. When conformal factors are not constant, we twist the sphere bundle to gain the pseudoconvexity. Then the algebraicity follows from the algebraicity theorem of Huang in the CR geometry. Different from the argument in the earlier works, the total geodesy of each factor does not directly follow from the properness because the codimension is arbitrary. By analyzing the real analytic subvariety carefully, we conclude that the factor is either a proper holomorphic rational map or a constant map. Lastly the total geodesy follows from a linearity criterion of Huang. On the other hand, we also derive the total geodesy for the local holomorphic isometries from the projective space to the product of projective spaces. In the second part, we give a proof for the convergence of a modified Kähler-Ricci flow. The flow is defined by Zhang on Kähler manifolds while the Kähler class along the evolution is varying. When the limit cohomology class is semi-positive, big and integer, the convergence of the flow is conjectured by Zhang and we confirm it by using the monotonicity of some energy functional. When the limit class is Kähler, the convergence is proven by Zhang and we give an alternative proof by also using the energy functional. As a corollary, the convergence provides the solution to the degenerate Monge-Ampère equation on the Calabi-Yau manifold. Meanwhile we take the opportunity to describe the Kähler-Ricci flow on singular varieties.