Recognizing the ${\rm G}_2$-horospherical manifold of Picard number 1 by varieties of minimal rational tangents

TL;DR

This paper characterizes the ${ m G}_2$-horospherical manifold of Picard number 1 uniquely by its variety of minimal rational tangents (VMRT), establishing a new recognition criterion based on differential geometric and deformation properties.

Contribution

It proves that the ${ m G}_2$-horospherical manifold is uniquely identified among Fano manifolds by its VMRT, extending methods previously used for symplectic Grassmannians.

Findings

${ m G}_2$-horospherical manifold is uniquely characterized by VMRT.

Established a recognition criterion for this manifold based on VMRT properties.

Developed a refined method involving positivity/negativity of vector bundles for this classification.

Abstract

Pasquier and Perrin discovered that the ${\rm G}_2$-horospherical manifold ${\bf X}$ of Picard number 1 can be realized as a smooth specialization of the rational homogeneous space parameterizing the lines on the 5-dimensional hyperquadric, in other words, it can be deformed nontrivially to the rational homogeneous space. We show that ${\bf X}$ is the only smooth projective variety with this property. This is obtained as a consequence of our main result that ${\bf X}$ can be recognized by its VMRT, namely, a Fano manifold of Picard number 1 is biregular to ${\bf X}$ if and only if its VMRT at a general point is projectively isomorphic to that of ${\bf X}$. We employ the method the authors developed to solve the corresponding problem for symplectic Grassmannians, which constructs a flat Cartan connection in a neighborhood of a general minimal rational curve. In adapting this method to ${\bf X}$, we need an intricate study of the positivity/negativity of vector bundles with respect to a family of rational curves, which is subtler than the case of symplectic Grassmannians because of the nature of the differential geometric structure on ${\bf X}$ arising from VMRT.