Abstract: We continue the study of the complex Monge-Ampère operator initiated by Prakhar. The homogeneous complex Monge--Ampère equation is the geodesic equation for a natural Riemannian metric on the space of Kähler metrics on a Kähler manifold. In this talk, we will compute the Levi-Civita connection and curvature on this infinite dimensional Riemannian manifold, and show that it is a symmetric space.
Abstract: When the line bundle is not positive, another useful tool to study the cohomologies of line bundles is Demailly's holomorphic Morse inequalities. We will extend Bergman kernel to (0,q)-forms valued in a holomorphic line bundle and discuss local holomorphic Morse inequalities due to Berman in his Ph.D thesis. Finally, if time permits, I will mention some works in my master thesis regarding this direction.
Abstract: When the line bundle is not positive, another useful tool to study the cohomologies of line bundles is Demailly's holomorphic Morse inequalities. We will extend Bergman kernel to (0,q)-forms valued in a holomorphic line bundle and discuss local holomorphic Morse inequalities due to Berman in his Ph.D thesis. Finally, if time permits, I will mention some works in my master thesis regarding this direction.
Abstract: The goal of the talk in these 2 weeks is to prove Calabi-Yau Theorem. This week, we will have a quick review on the definition and properties of Kähler manifold. Then we will formulate Calabi-Yau Theorem into a PDE problem and we will show the theorem using the continuity method. We will outline the several key steps of the proof and leave the technical details in next week. The main reference is chapter 1 -3 of An Introduction to Extremal Kähler Metrics written by Gabor Szekelyhidi.
Abstract: This week, we continue the proof of Calabi-Yau Theorem. In particular, we will show the remaining proof of openness and closeness from last week and Yau's C^0 estimate. If time permits, we can discuss its application in Kähler-Ricci Flow.
Abstract: In this talk, we continue discussion on stability for holomorphic vector bundles initiated by Sze-Hong in the last semester. We will discuss Donaldson-Uhlenbeck-Yau theorem/Kobayashi-Hitchin correspondence, which generalizes Narasimhan-Seshadri theorem to higher dimensions. Following Yum-Tong Siu’s note, we will present Donaldson’s heat flow proof to prove that any stable bundle on projective manifolds admits Hermitian-Einstein metric.
Abstract: After recalling the pluripotential theory on Open sets in Cn discussed last semester, I'll describe the pluripotential theory on compact Kahler manifolds. I'll set up the problem of solving the complex Monge Ampere equation using the variational method.