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theses

In this thesis, we study three problems related to Complex Monge-Amp`ere equations. After the introduction and preliminary, In chapter 3, we study K ̈ahler Ricci flow on Fano bundle, with finite time singularity. we show that under the suitable assumption on the initial and ending K ̈ahler class, the evolving K ̈ahler metrics along K ̈ahler Ricci flow have uniform diameter bound and moreover, if we assume the fiber of Fano bundle is Pn or Mm,k, the evolving metric will converge to a K ̈ahler metric on the base of the Fano bundle in Gromov-Hausdorff sense, which generalizes the result of Song-Szekelyhidi- Weinkove [103] who study the K ̈ahler Ricci flow on projective bundle.

In chapter 4, based on Kolodziej’s fundamental result on C0 estimate of complex Monge-Amp`ere equation, we study the geometric property of complex manifolds cou- pled with a family of K ̈ahler metrics which come from solutions of a family of complex Monge-Amp`ere equations. As a application, on a minimal K ̈ahler manifold with inter- mediate Kodaira dimension, we obtain uniform diameter bound of a family of collapsing K ̈ahler metrics whose K ̈ahler class is small perturbation of the canonical class. This is our first attempt to understand canonical metric on complex manifold with nef canon- ical class.
In chapter 5, we further study degeneration of K ̈ahler-Einstein metrics with negative curvature on canonical polarized complex manifold. For this purpose, we construct complete K ̈ahler-Einstein metric near isolated log canonical singularity through two different methods and for those log canonical singularity coupled with a model metric satisfying bounded geometry property roughly, we prove a rigidity result concerning complete K ̈ahler-Einsteins near the singularity.

ETD_10665

Ph.D.

Includes bibliographical references

doi:10.7282/t3-ftz8-ay18

ETD doctoral

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