Positivity in Complex Geometry

本课题研究复几何中正性刻画的基本问题。复数几何是几何学的一个分支，它涉及复数域上定义的对象。这种正性意味着存在大量被称为全纯截面的整体对象，它们在几何学中扮演着核心角色。这项研究将发展和引入不同领域的技术，包括微分几何、代数几何和非线性偏微分方程组。其中一些答案将阐明复杂的几何结构和代数结构之间的深层联系。该建议还将为研究生和初级研究人员产生适当的研究问题。复几何中正则丛的正性引起了复几何、非线性偏微分方程式和代数几何之间有趣的联系。PI将进行三个研究项目，以更好地从使用全纯曲率、Kahler-Einstein度量和复杂的Monge-Ampere方程的微分几何方法来理解正则丛的正性。还将研究一类推广复Monge-Ampere方程的完全非线性方程。第一个项目将利用全纯曲率、双曲性和Bergman度量来研究正则丛的正性。第二个项目将研究正对数正则丛上正则度量的渐近展开。第三个项目试图建立一些完全非线性方程的Liouville-型定理。

This project studies fundamental problems on the characterization of positivity in complex geometry. Complex geometry is a branch of geometry that concerns objects defined over the complex number field. The positivity implies the existence of plenty of global objects called holomorphic sections, which play a central role in geometry. The research will develop and bring in techniques from different fields including differential geometry, algebraic geometry, and nonlinear partial differential equations. Some of the answers will shed light on the deep connection between the complex geometric structure and algebraic structure. The proposal will also generate appropriate research problems for graduate students and junior researchers.Positivity of canonical bundles in complex geometry gives rise to intriguing connections between complex geometry, nonlinear partial differential equations, and algebraic geometry. The PI will undertake three research projects to better understand the positivity of canonical bundles from a differential geometric approach using holomorphic curvature, Kahler-Einstein metrics, and the complex Monge-Ampere equation. A class of fully nonlinear equations generalizing the complex Monge-Ampere equation will be also investigated. The first project will study the positivity of canonical bundles by holomorphic curvature, hyperbolicity, and the Bergman metric. The second project will investigate the asymptotic expansion of the canonical metric on the positive logarithmic canonical bundle. The third project seeks to establish a Liouville-type theorem for some fully-nonlinear equations.