Fubini-Griffiths-Harris rigidity and Lie algebra cohomology

Abstract

We prove a rigidity theorem for represented semi-simple Lie groups. 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 (with the exception of $\mathfrak{g} = \mathfrak{a}_1$).

In contrast, we show that the adjoint variety of $SL_3\mathbb{C}$ and the Segre product $Seg(\mathbb{P}^1 \times \mathbb{P}^n$) are flexible at order two. In the $SL_3\mathbb{C}$ 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.