2007/07/23 by J. M. Landsberg, Landsberg, J. M., Colleen Robles +2
Mathematics · #17B56 #51N35 #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #Differential Geometry (math.DG) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT) #math.AG #math.DG #math.RT #msc:17B56 #msc:51N35
paper · pdf · doi:10.48550/arxiv.0707.3410
v.1: 25 pages. v.2: The exposition has been improved and the language of filtered EDS used
openalex publication_date 2007/07/23 · arxiv created 2008/02/06 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove a general extrinsic rigidity theorem for homogeneous varieties in \mathbbCPN. The theorem is used to show that the adjoint variety of a complex simple Lie algebra \mathfrakg (the unique minimal G orbit in ℙ\mathfrakg) is extrinsically rigid to third order. In contrast, we show that the adjoint variety of SL3ℂ, and the Segre product Seg(ℙ1× ℙn), both varieties with osculating sequences of length two, are flexible at order two. In the SL3ℂ 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.