Fubini-Griffiths-Harris rigidity and Lie algebra cohomology

TL;DR

This paper establishes a general extrinsic rigidity theorem for homogeneous varieties in complex projective space, demonstrating rigidity for certain adjoint varieties and flexibility for others, using Lie algebra cohomology and differential systems.

Contribution

It introduces a broad extrinsic rigidity theorem for homogeneous varieties and applies it to specific cases, extending existing geometric and cohomological methods.

Findings

Adjoint variety of a complex simple Lie algebra is rigid to third order.

Adjoint variety of SL_3(C) and Segre products are flexible at second order.

The paper extends machinery linking Lie algebra cohomology with geometric rigidity and flexibility.

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 $SL_3\mathbb{C}$, and the Segre product $\mathit{Seg}(\mathbb{P}^1\times \mathbb{P}^n)$, both varieties with osculating sequences of length two, 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.