Fubini-Griffiths-Harris rigidity and Lie algebra cohomology

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dc.creatorLandsberg, J. M.
dc.creatorRobles, C.
dc.date2007-07-23
dc.date2008-02-06
dc.date.accessioned2026-07-07T09:18:42Z
dc.date.available2026-07-07T09:18:42Z
dc.descriptionWe 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.
dc.descriptionv.1: 25 pages. v.2: The exposition has been improved and the language of filtered EDS used
dc.identifierhttps://arxiv.org/abs/0707.3410
dc.identifierhttp://arxiv.org/abs/0707.3410
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/154121
dc.subjectDifferential Geometry
dc.subjectAlgebraic Geometry
dc.subjectRepresentation Theory
dc.subject51N35; 17B56
dc.titleFubini-Griffiths-Harris rigidity and Lie algebra cohomology
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