Complex Finsler Geometry and Related Topics

PI建议在芬斯勒几何中的概念与代数几何中的概念之间建立对应关系。更具体地说，我们想找到与复芬斯勒几何中的微分几何概念相对应的代数概念：（a）全纯双截曲率，（B）里奇曲率，（c）全纯截曲率，（d）标量曲率，（e）Bochner张量（Weyl共形张量的复模拟）。芬斯勒方法将适用于某些相干层，而不仅仅是埃尔米特几何的经典理论中的向量丛。它还允许我们扩展理论从复杂的品种定义在其他领域，如p-adic数fields.I的动机部分由经典理论的内在度量（如小林和Caratheodory度量），在一般情况下，芬斯勒，而不是厄米特。另一个动机来自于基于芬斯勒-爱因斯坦-洛伦兹度规的广义相对论的最新发展。因此，一个严格的芬斯勒几何方法是必不可少的理解这些指标。在这个理论中，时间锥不是圆形的（两个圆锥，一个在另一个的顶部），而是金字塔的形状（两个金字塔，一个在另一个的顶部），几何形状要复杂得多。对Finsler Bochner-Weyl张量的理解将是非常有帮助的，因为它与Twistor理论和Yang-Mill自对偶有关。

The PI proposes to establish a correspondence between concepts in FINSLER GEOMETRY with concepts in ALGEBRAIC GEOMETRY. To be more specific we want to find the algebraic concepts which correspond to the differential geometric concepts in Complex Finsler Geometry:(a) Holomorphic bisectional curvature,(b) Ricci curvature,(c) Holomorphic sectional curvature,(d) Scalar curvature,(e) Bochner tensor (the complex analogue of the Weyl conformal tensor).The Finsler approach would work for certain coherent sheaves rather than just vector bundles as in the classical theory of Hermitian geometry. It also allows us to extend the theory from complex to varieties defined over other fields such as the p-adic number fields.I am motivated in part by the classical theory of intrinsic metrics (such as the Kobayashi and Caratheodory metrics) which are, in general, Finslerian rather than Hermitian. Another motiviation comes from the recent development in general relativity based on Finsler-Einstein-Lorentze metrics. Thus a rigorous Finsler geometric approach is essential to understand these metrics. In this theory the time cone is not circular (two cones, one on top of the other) but in the shape of pyramids (two pyramids, one on top of the other) and the geometry is much more intricate. The understanding of the Finsler Bochner-Weyl tensor would be very helpful as it is related to Twistor theory and the Yang-Mill self-duality.