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: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=740118
• 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.

该研究计划在于复杂的几何形状领域。复杂的几何形状是Riemannian几何形状扩展到复杂的世界，其中关键的几何对象（歧管，生存在歧管上方的束）具有全态过渡函数。我的研究计划涉及对某些居住在复杂歧管或全体形态矢量束上的指标的研究，并具有特殊的曲率特性。这些指标是非线性偏微分方程（PDE）的先验解决方案，其存在的问题在大多数时候非常微妙，需要不同技术的混合物（全球分析，多能理论，复杂的微分几何，代数几何，代数几何，几何学，几何不变理论）。一个典型的例子是爱因斯坦度量的一般相对论。由于它们与其他领域的关系（Symplex几何，弦理论，数学物理学，拓扑，非架构的几何形状），对这些指标的研究是加拿大和国外的一个非常活跃的研究主题。例如，请提一说，在过去的几十年中，各种PDE解决方案的模量空间对它们所处的基本几何对象的分类非常有益。我的程序中描述的特定对象地址以下牢固连接的方向，无论是从几何角度还是使用的技术：（i）对于平滑的多种流形，我希望几何量化能够提供一个新的完整量化，以提供新的完整性洞察力，可以为寄居者方程式提供了杂物的连接，从而提供了远处的远程连接（也可以恢复遗传）。这个话题。从一般的角度来看，我计划使用几何量化实施的方法应该足够强大，可以长期解决概括，以“装饰”捆绑包（不一定是平滑）变化。 （ii）在作为矢量束的投影限制的统治歧管上，在基础捆绑包上的赫米尔米尼 - 因斯坦公制的存在与至少在统治的歧管上的恒定标量曲率度量（Einstein Metric的概括）的存在有关，至少在捆绑包上定义了复杂的弯曲曲线。我的目标是证明这种关系的奇异指标的扩展，提供了对数版本Yau-Tian-Donaldson的对数版本的证据，Yau-Tian-Donaldson猜想是该领域的中心猜想。我还打算研究当人们考虑一个生活在更高维歧管上的捆绑包的项目时，正在发生的事情。该研究计划包括对几个HQP进行培训，这些HQP将获得广泛的知识。从一般的角度来看，该研究计划倾向于使用完全技术的协同作用加深对某些基本几何对象的理解。