1. Recognizing the G2-horospherical Manifold of Picard Number 1 by Varieties of Minimal Rational Tangents.
- Author
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Hwang, Jun-Muk and Li, Qifeng
- Abstract
Pasquier and Perrin discovered that the G
2 -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. [ABSTRACT FROM AUTHOR]- Published
- 2024
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