Exterior differential systems, Lie algebra cohomology, and the rigidity of homogenous varieties

TL;DR

This paper introduces exterior differential systems (EDS) and discusses recent advances in using EDS to study the rigidity of rational homogeneous varieties, connecting differential geometry, Lie algebra cohomology, and PDE techniques.

Contribution

It provides an accessible introduction to EDS and presents new results on the rigidity of homogeneous varieties using advanced EDS methods.

Findings

Enhanced understanding of the rigidity of rational homogeneous varieties.

Development of EDS techniques for studying geometric rigidity.

Connections established between Lie algebra cohomology and geometric properties.

Abstract

These are expository notes from the 2008 Srni Winter School. They have two purposes: (1) to give a quick introduction to exterior differential systems (EDS), which is a collection of techniques for determining local existence to systems of partial differential equations, and (2) to give an exposition of recent work (joint with C. Robles) on the study of the Fubini-Griffiths-Harris rigidity of rational homogeneous varieties, which also involves an advance in the EDS technology.