Schubert varieties and cycle spaces

Complex geometric properties of the orbits of a non-compact real form $G_0$ in a flag manifold $Z=G/Q$ of a complex semi-simple groups $G=G_0^\mathbb C$ are studied. Schubert varieties are used to construct a complex submanifold with optimal slice properties in any given $G_0$-orbit. For an open $G_0$- orbit $D$, given $p$ in the boundary of $D$, a variety $Y\setminus D$ containing $p$ with maximal dimension with respect to the compact cycles in $D$ is constructed. The method of incidence varieties then yields information on the complex geometry of the associated cycle space. In particular, holomorphic convexity is verified and in the Hermitian case a fine classification is obtained.