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Fubini-Griffiths-Harris rigidity and Lie algebra cohomology | | J.M. Landsberg
; C. Robles
; | | Date: |
23 Jul 2007 | | 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 $SL_3mathbb{C}$, and the
Segre product $mathit{Seg}(mathbb{P}^1 imes mathbb{P}^n)$, both varieties
with osculating sequences of length two, are flexible at order two. In the
$SL_3mathbb{C}$ 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. | | Source: | arXiv, 0707.3410 | | Services: | Forum | Review | PDF | Favorites | |
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