Recognizing the G 2-horospherical Manifold of Picard Number 1 by Varieties of Minimal Rational Tangents

Abstract

Pasquier and Perrin discovered that the G2-horospherical manifold 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 X is the only smooth projective variety with this property. This is obtained as a consequence of our main result that X can be recognized by its VMRT, namely, a Fano manifold of Picard number 1 is biregular to X if and only if its VMRT at a general point is projectively isomorphic to that of 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 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 X arising from VMRT. © 2023, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.