Jun-Muk Hwang

Title: Deformation of the space of lines on the 5-dimensional hyperquadric.

Abstract: Let $F^5$ be the space of lines on the 5-dimensional hyperquadric $Q^5 \subset {\bf P}^6$. $F^5$ is a 7-dimensional homogeneous projective manifold. We show that a projective manifold which arises as a deformation of $F^5$ is biregular

to either $F^5$ itself or the $G_2$-horospherical variety $X^5$ studied by Pasquier-Perrin. The key point of the proof is to show that a 7-dimensional uniruled projective manifold of

Picard number 1 with the variety of minimal rational tangents isomorphic to a certain Hirzebruch surface is biregular to $X^5$. A main new ingredient in the proof is a study of the Cartanian geometry of the geometric structure determined by such a Hirzebruch surface: the construction of a Cartan connection and the investigation of its curvature. This geometric structure is associated to a non-reductive graded Lie algebra and has not been studied classically.