On the topology of minimal orbits in complex flag manifolds

Abstract

We compute the Euler-Poincaré characteristic of the homogeneous compact manifolds that can be described as minimal orbits for the action of a real form in a complex flag manifold.

Complex vector fields and hypoelliptic partial differential operators

“…In fact, in §2 we prove that, at points that are generic for Z(M ) (in a sense made precise in Definition 2.15), it is equivalent to a condition (2.32), that involves semidefinite generalized Levi forms attached to Z(M ) and their kernels. This quite explicit formulation was suggested by specific examples from [1]. However, conditions (1.21) and (2.32) apply to more general contexts than CR geometry.…”

“…It is connected and has minimal dimension. Minimal orbits have been studied, from the point of view of CR geometry, in [1]. If (g, q) is the CR algebra of a minimal orbit M , then g o contains a maximally vectorial Cartan subalgebra h of g. This choice of h is equivalent to the fact that R im = R • , i.e.…”

“…The study of their CR geometry has been already started in [1,2], to which we refer for the complete explanation of many details.…”

“…After considering general homogeneous CR manifolds in §6, in §7 we classify all minimal orbits of real forms G of G C -homogeneous flag manifolds (see e.g. [24,1,2]) that enjoy condition (1.21). In [1, §13], together with Prof. Medori, two of the authors gave the complete classification of the essentially pseudoconcave minimal orbits.…”

“…[1]), i.e. by systems Φ ⊂ B, where B are the simple roots of a positive system R + satisfying (7.19).…”