Geometric quantization and metrics with special curvature properties

几何量化和具有特殊曲率特性的度量

• 批准号: RGPIN-2020-04683
• 负责人: Keller, Julien
• 金额: $ 1.75万
• 依托单位: Université du Québec à Montréal
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759324
• 关键词: Geometric; quantization; metrics; special; curvature

This research program lies in the area of complex geometry. Complex geometry is the extension of Riemannian geometry to the complex world where the key geometric objects (manifolds, bundles that live above manifolds) have holomorphic transition functions.  My research program deals with the study of certain metrics that live either on complex manifolds or on holomorphic vector bundles and have special curvature properties. These metrics are transcendental solutions of non linear partial differential equations (PDE) and the question of their existence is most of the time very subtle and requires a mixture of different technologies (global analysis, pluripotential theory, complex differential geometry, algebraic geometry, geometric invariant theory). A typical example is the Einstein metric in General Relativity. Due to their relationship with other fields (symplectic geometry, string theory, mathematical physics, topology, non-Archimedean geometry.), the study of these metrics is a very active research subject in Canada and abroad. For example, let's mention that during the last decades, the construction of moduli spaces of solutions of various PDE has been very fruitful for classifications of the underlying geometric objects on which they live. The specific objectives described in my program address the following strongly connected directions, both from a geometrical perspective and the techniques used: (I) For holomorphic vector bundles over a smooth manifold, I expect geometric quantization to provide a new complementary insight on the metrics which solve the Hermitian-Einstein equation (also called Hermitian Yang-Mills equation for the Chern connection in Physics), retrieving classical and deep results on this topic. From a general point of view, the method I plan to implement with geometric quantization should be robust enough to tackle generalizations, in the long term, to "decorated" bundles over (not necessarily smooth) varieties. (II) On a ruled manifold given as the projectivisation of a vector bundle, the existence of a Hermitian-Einstein metric on the underlying bundle is related to the existence of a constant scalar curvature metric (a generalization of the Einstein metric) on the ruled manifold, at least when the bundle is defined over a complex curve. I aim to prove an extension of this relation for singular metrics, providing evidence of a logarithmic version of the Yau-Tian-Donaldson conjecture, a central conjecture in the field. I also intend to study what is happening when one is considering the projectivisation of a bundle that lives over higher dimensional manifolds. The research program includes the training of several HQP that will acquire a wide spectrum of knowledge. From a general perspective, this research program tends to deepen the understanding of certain fundamental geometric objects using the synergy of complementary techniques.

这个研究项目属于复杂几何学领域。复几何是黎曼几何在复世界中的延伸，其中的关键几何对象（流形，流形上的丛）具有全纯转移函数。我的研究项目涉及研究某些度量，这些度量既可以存在于复流形上，也可以存在于全纯向量丛上，并且具有特殊的曲率性质。这些度量是非线性偏微分方程（PDE）的超越解，它们的存在性问题在大多数情况下是非常微妙的，需要不同技术的混合（全局分析，多能理论，复微分几何，代数几何，几何不变理论）。一个典型的例子是广义相对论中的爱因斯坦度量。由于它们与其他领域（辛几何，弦理论，数学物理，拓扑学，非阿基米德几何）的关系，对这些度量的研究在加拿大和国外是一个非常活跃的研究课题。例如，让我们提到，在过去的几十年中，各种PDE的解决方案的模空间的建设已经非常富有成效的分类的基础几何对象，他们生活。在我的程序中描述的具体目标解决了以下强烈连接的方向，无论是从几何角度还是所使用的技术：（I）对于光滑流形上的全纯向量丛，我期望几何量子化能为解决Hermitian-Einstein方程的度量提供一个新的补充见解。（也称为Hermitian Yang-Mills方程的物理学陈连接），检索经典和深入的结果在这个话题上。从一般的角度来看，我计划用几何量化实现的方法应该足够健壮，从长远来看，可以解决推广问题，在（不一定是光滑的）品种上“装饰”束。(II)在作为向量丛的投影化给出的直纹流形上，基本丛上的厄米-爱因斯坦度量的存在性与直纹流形上的常数标量曲率度量（爱因斯坦度量的推广）的存在性有关，至少当丛定义在复曲线上时是这样。我的目标是证明这种关系的奇异度量的扩展，提供证据的对数版本的姚田-唐纳森猜想，在该领域的中心猜想。我还打算研究当人们考虑在高维流形上的丛的投影时会发生什么。该研究计划包括培训几名HQP，他们将获得广泛的知识。从一般的角度来看，这个研究计划倾向于使用互补技术的协同作用加深对某些基本几何对象的理解。