Geometric Structures and Substructures on Uniruled Projective Manifolds

Abstract

In a series of works on uniruled projective manifolds starting in the late 1990’s, Jun-Muk Hwang and the author have developed the basics of a geometric theory of uniruled projective manifolds arising from the study of varieties of minimal rational tangents (VMRTs), i.e., the collection at a general point of tangents to minimal rational curves passing through the point. From its onset, our theory is a cross-over between algebraic geometry and differential geometry. While we deal with problems in algebraic geometry, the heart of our perspective is differential-geometric in nature, revolving around foliations, G-structures, differential systems, etc. and dealing with various issues relating to connections, curvature and integrability.The current article is written with the aim of highlighting certain aspects in the geometric theory of VMRTs revolving around the theme of analytic continuation of geometric structures and substructures. For the parts of the article where adequate exposition already exists, we recall fundamental elements and results in the theory essential for the understanding of more recent development and provide occasional examples for illustration. The presentation will be more systematic on sub-VMRT structures since the latter topic is relatively new. We will discuss various perspectives concerning sub-VMRT structures, and indicate how the subject has intimate links with other areas of mathematics including several complex variables, local differential geometry and Kähler geometry.