Deformation rigidity of the rational homogeneous space associated to a long simple root

Abstract

As a continuation of our previous works we study the conjecture on the rigidity under Kähler deformation of the complex structure of rational homogeneous spaces G / P of Picard number 1, confirming its validity whenever G / P is associated to a long simple root. For these rational homogeneous spaces the minimal G-invariant holomorphic distribution D is spanned by varieties of minimal rational tangents, and, excepting the symmetric and the contact cases, the complex structure of G / P is completely determined by the nilpotent symbol algebra of the weak derived differential system of D. The problem is reduced, in a sense, to the invariance of this nilpotent symbol algebra under Kähler deformation. In our earlier works in relation to the question of the integrability of distributions spanned by varieties of minimal rational tangents we have established identities on Lie brackets using integral surfaces arising from pencils of rational curves. In the case on hand, at a point oε G / P we prove that the nilpotent symbol algebra at o is nothing other than the universal Lie algebra generated by Do subject to these identities on Lie brackets, by verifying that they correspond to finiteness condition in the Serre presentation of the simple Lie algebra G. © 2002 Éditions scientifiques et médicales Elsevier SAS.