2022/12/19 by Hwang, Jun-Muk, Li, Qifeng
#14M17 #32G05 #53C15 #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2212.09226
Pasquier and Perrin discovered that the \rm G2-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.