Geometric structures on uniruled projective manifolds defined by their varieties of minimal rational tangents

Abstract

In a joint research programme with Jun-Muk Hwang we have been investigating geometric structures on uniruled projective manifolds, especially Fano manifolds of Picard number 1, defined by varieties of minimal rational tangents associated to moduli spaces of minimal rational curves. In this article we outline a heuristic picture of the geometry of Fano manifolds of Picard number 1 with non-linear varieties of minimal rational tangents, taking as hints from prototypical examples such as those from holomorphic conformal structures. On an open set in the complex topology the local geometric structure associated to varieties of minimal rational tangents is equivalently given by families of local holomorphic curves marked at a variable base point satisfying certain compatibility conditions. Differential-geometric notions such as (null) geodesics, curvature and parallel transport are a source of inspiration in our study. Formulation of problems suggested by this heuristic analogy and their solutions, sometimes in a very general context and at other times applicable only to special classes of Fano manifolds, have led to resolutions of a series of well-known problems in Algebraic Geometry. © Astérisque 322.