Integral varieties of the canonical cone structure on \(G/P\)
For every point \(x\) in a compact symmetric Hermitian space \(M=G/P\), the associated cone \(K_x\) in \(T_xM\) is by definition the cone of highest weight vectors under the action of the reductive part of the parabolic subgroup \(P\). It is known that the cone \(K_x\) coincides with the cone of the vectors tangent to the lines in \(G/P\) passing through \(x\). \(K\) is the union of the \(K_x\). An integral subvariety \(X\) of \(K\) is a subvariety of \(M\) such that the tangent bundle for the smooth part is included in \(K\). \(M\) can also be imbedded equivariantly in some canonical projective space. In view of this, \(K_x\) coincides with \(M\cap T_xM\). Under this embedding, the authors give an answer to the question: When is an integral subvariety contained in a linear space on \(M\)? The authors prove that the answer is positive when \(G\) is not of type \(B_n\) or \(C_n\) and when the dimension of \(X\) is bigger than some number \(d\) depending on \(M\).
scientific article; zbMATH DE number 2123986