Positivity in Complex Geometry

本课题研究复杂几何中正性表征的基本问题。复几何是几何学的一个分支，涉及在复数域上定义的对象。正性意味着存在大量被称为全纯截面的整体对象，它们在几何中起着中心作用。该研究将发展并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。其中一些答案将揭示复杂几何结构与代数结构之间的深层联系。该提案还将为研究生和初级研究人员产生适当的研究问题。复杂几何中正则束的正性引起了复杂几何、非线性偏微分方程和代数几何之间的有趣联系。PI将承担三个研究项目，以更好地理解正则束的正性，从微分几何方法使用全纯曲率，Kahler-Einstein度量，和复杂的蒙日-安培方程。本文还研究了一类推广复蒙日-安培方程的全非线性方程。第一个项目将通过全纯曲率、双曲度和伯格曼度规研究正则束的正性。第二个项目将研究正则度量在正对数正则束上的渐近展开式。第三个项目寻求建立一些全非线性方程的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.