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

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. Date: November 15, 2018. J.M. LANDSBERG 6.4. An easier path to rigidity? 17 7. Osculating gradings and root gradings 17 7.1. The osculating filtration 18 7.2. The root grading 18 7.3. Examples of tangent spaces and osculating filtrations of homogeneous varieties 19 8. Lie algebra cohomology and Kostant's theory 22 9. From the Fubini EDS to Filtered EDS 22 9.1. Problem 1: Osculating v.s. root gradings 22 9.2. Problem 2: Even the systems defined by the root grading do not lead to Lie algebra cohomology 23 9.3. The Fix for problem 2: Filtered EDS 24 10. Open questions and problems 26 References 26