Integral varieties of the canonical cone structure on <Emphasis Type="Italic">G</Emphasis>/<Emphasis Type="Italic">P</Emphasis> |
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Authors: | Email author" target="_blank">Insong?ChoeEmail author Jaehyun?Hong |
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Institution: | (1) Korea Institute for Advanced Study, 207-43 Cheongnyangni 2-dong, Dongdaemun-gu, Seoul, 130-722, Korea |
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Abstract: | The canonical cone structure on a compact Hermitian symmetric space G/P is the fiber bundle
where
is the cone of the highest weight vectors under the action of the reductive part of P. It is known that the cone
coincides with the cone
of the vectors tangent to the lines in G/P passing through x, when we consider G/P as a projective variety under its homogeneous embedding into the projective space
of the irreducible representation space V
of G with highest weight associated to P. A subvariety X of G/P is said to be an integral variety of
at all smooth points xG/P. Equivalently, an integral variety of
is a subvariety of G/P whose embedded projective tangent space at each smooth point is a linear space
We prove a kind of rigidity of the integral varieties under some dimension condition. After making a uniform setting to study the problem, we apply the theory of Lie algebra cohomology as a main tool. Finally we show that the dimension condition is necessary by constructing counterexamples. |
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Keywords: | 53Cxx 32Mxx 17Bxx |
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