Fubini-Griffiths-Harris rigidity and Lie algebra cohomology

Abstract

We prove a general extrinsic rigidity theorem for homogeneous varieties in ${\mathbb{CP}}^N$. The theorem is used to show that the adjoint variety of a complex simple Lie algebra $\mathfrak{g}$ (the unique minimal G orbit in $\mathbb{P}\mathfrak{g}$) is extrinsically rigid to third order. In contrast, we show that the adjoint variety of $S{L}_3\mathbb{C}$, and the Segre product $Seg({\mathbb{P}}^1\times {\mathbb{P}}^n)$, both varieties with osculating sequences of length two, are flexible at order two. In the $S{L}_3\mathbb{C}$ example we discuss the relationship between the extrinsic projective geometry and the intrinsic path geometry. We extend machinery developed by Hwang and Yamaguchi, Se-ashi, Tanaka and others to reduce the proof of the general theorem to a Lie algebra cohomology calculation. The proofs of the flexibility statements use exterior differential systems techniques.

v.1: 25 pages. v.2: The exposition has been improved and the language of filtered EDS used