Geometry of varieties of minimal rational tangents

Summary

We present the theory of varieties of minimal rational tangents (VMRT), with an emphasis on its own structural aspect, rather than applications to concrete problems in algebraic geometry. Our point of view is based on differential geometry, in particular, Cartan's method of equivalence. We explain various aspects of the theory, starting with the relevant basic concepts in differential geometry and then relating them to VMRT. Several open problems are proposed, which are natural from the view point of understanding the geometry of VMRT itself.

1. Introduction

The concept of varieties of minimal rational tangents (VMRT) on uniruled projective manifolds first appeared as a tool to study the deformation of Hermitian symmetric spaces [Hwang and Mok 1998]. For many classical examples of uniruled manifolds, VMRT is a very natural geometric object associated to low degree rational curves, and as such, it had been studied and used long before its formal definition appeared in that reference. At a more conceptual level, namely, as a tool to investigate unknown varieties, it had been already used in [Mok 1988] for manifolds with nonnegative curvature. However, in the context of that work, its very special relation with the curvature property of the Kähler metric somewhat overshadowed its role as an algebro-geometric object, so it had not been considered for general uniruled manifolds. Thus it is fair to say that the concept as an independent geometric object defined on uniruled projective manifolds really originated from [Hwang and Mok 1998]. Shortly after this formal debut, numerous examples of its applications to classical problems of algebraic geometry were discovered. In the early MSRI survey [Hwang and Mok 1999], written only a couple of years after the first discovery, one can already find a substantial list of problems in a wide range of topics, which can be solved by the help of VMRT.

Since the beginning VMRT has been studied exclusively in relation with some classical problems, namely, problems which do not involve VMRT itself explicitly. In particular, this is the case for most of my collaboration with N. Mok. In other words, VMRT has mostly served as a tool to study uniruled manifolds. However, after more than a decade's service, I believe it is time to give due recognition and it is not unreasonable to start to regard VMRT itself as a central object of research.