On the Kahler Ricci flow, positive curvature in Hermitian geometry and non-compact Calabi-Yau metrics

In this thesis, we study three problems in complex geometry. In the first part, we study the behavior of the Kahler-Ricci flow on complete non-compact manifolds with negative holomorphic curvature. We show that Kahler-Ricci flow converges to a Kahler-Einstein metric when the initial manifold admits a suitable exhaustion function, thus improving upon a result of D. Wu and S.T. Yau. These results are partly obtained in joint work with S. Huang, M.-C. Lee and L.-F. Tam. In the second part of this thesis, we introduce a new Kodaira-Bochner type formula for closed (1, 1)-form in non-Kahler geometry. Based on this new formula, We propose a new curvature positivity condition in non-Kahler manifolds and proved a strong rigidity type theorem for manifolds satisfying this curvature positivity condition. We also find interesting examples non-Kahler manifolds satisfying the curvature positivity condition in a class of manifolds called Vaisman manifolds. In the third part of this thesis, we study the degenerations of asymptotically conical Calabi-Yau manifolds as the Kahler class degenerates to a non-Kahler class. Under suitable hypothesis, we prove the convergence of asymptotically conical Calabi-Yau metrics to a singular asymptotically conical Calabi-Yau current with compactly supported singularities. Using this, we construct singular asymptotically conical Calabi-Yau metrics on non-compact singular varieties and identify the topology of these singular metrics with the singular variety. We also give some interpretations of these asymptotically conical Calabi-Yau metrics from the point of view of physics. These results are obtained in joint work with T. Collins and B. Guo.