Partial compactification of metabelian Lie groups with prescribed varieties of minimal rational tangents

TL;DR

This paper characterizes which isotropic varieties can serve as VMRTs of minimal rational curves tangent to distributions on complex manifolds, constructing such manifolds as partial compactifications of metabelian groups.

Contribution

It provides a converse to the known conditions on VMRTs, showing any isotropic variety can be realized as a VMRT on a specially constructed complex manifold.

Findings

Any smooth isotropic variety can be realized as a VMRT.

Constructs complex manifolds as partial compactifications of metabelian groups.

Establishes a converse to the isotropic VMRT condition.

Abstract

We study minimal rational curves on a complex manifold that are tangent to a distribution. In this setting, the variety of minimal rational tangents (VMRT) has to be isotropic with respect to the Levi tensor of the distribution. Our main result is a converse of this: any smooth projective variety isotropic with respect to a vector-valued anti-symmetric form can be realized as VMRT of minimal rational curves tangent to a distribution on a complex manifold. The complex manifold is constructed as a partial equivariant compactification of a metabelian group, which is a result of independent interest.