Existence of rational curves on algebraic varieties, minimal rational tangents, and applications

Abstract

This survey paper discusses some of the recent progress in the study of rational curves on algebraic varieties. It was written for the survey volume of the priority programme "Global Methods in Complex Geometry", supported by the DFG. To start, we discuss existence results for rational curves on foliated manifolds. We construct the tangent morphism, define the variety of minimal rational tangents (VMRT) and name a number of results that show that many of the geometry properies of a uniruled variety are encoded in the projective geometry of the VMRT. The results are then applied in two different settings. For one, we discuss the geometry of chains of rational curves on uniruled varieties, show how the length of a variety can be determined from the VMRT and discuss the concrete example of the moduli space of vector bundles on a curve. On the other hand, the existence results for rational curves can be also be used to study manifolds which are known NOT to contain rational curves. We employ this method to study deformations of morphisms. Finally, we look at families of varieties of canonically polarized manifolds over complex surfaces, and use non-existence results for rational curves on the base to relate the variation of the family with the logarithmic Kodaira dimension of the base.