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.
