Geometric quantization and metrics with special curvature properties

这个研究项目属于复杂几何领域。复几何是黎曼几何向复世界的扩展，在复世界中，关键几何对象（流形，流形之上的束）具有全纯过渡函数。我的研究项目是研究某些存在于复流形或全纯矢量束上并具有特殊曲率性质的度量。这些度量是非线性偏微分方程（PDE）的超越解，它们存在的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、几何不变量理论）。一个典型的例子是广义相对论中的爱因斯坦度规。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，这些度量的研究在加拿大和国外都是一个非常活跃的研究课题。例如，让我们提一下，在过去的几十年里，各种PDE解的模空间的构造对于它们所处的底层几何对象的分类非常有成效。

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.