Qaisar Latif

• Flag domains $D$ are by definition open orbits of real forms $G_0$ of complex semisimple groups $G$ in flag manifolds $Z=G/Q$. Every maximal compact subgroup $K_0$ has a unique orbit $C_0$ in $D$ which is a complex submanifold. Since $C_0$ can be moved with the group action, it is rare that $D$ possesses non-constant holomorphic functions. For example, if $D$ is cycle connected i.e., any two points can be connected by a chain of translates of $C_0$, then $\mathcal{O}(D)\cong \mathbb{C}$. In a 2010 paper of A. Huckleberry entitled "Remarks on homogeneous complex manifolds satisfying Levi conditions" it is shown $\mathcal{O}(D)\not\cong \mathbb{C}$, absence of cycle connectivity and pseudoconvexity (in the sense of plurisubharmonic exhaustions) are equivalent conditions. There it is conjectured that if one of these equivalent conditions is not fulfilled, then it is pseudoconcave in the sense of A. Andreotti. This conjecture is proved here. Rough estimates of the degree of pseudoconcavity are proved in general and explicit computations of various cycle- and curve-connectivity properties of $SU(p,p')$-flag domains are carried out.