Hermitian Symmetric Spaces, Cycle Spaces, and the Barlet–Koziarz Intersection Method for Construction of Holomorphic Functions

Under certain conditions, a recent method of Barlet and Koziarz \cite{BK} constructs enough holomorphic functions to give a direct proof of the Stein condition for a cycle space. Here we verify those conditions for open $G_0$–orbits on $X$, where $G_0$ is the group of a bounded symmetric domain and $X$ is its compact dual viewed as a flag quotient manifold of the complexification $G$ of $G_0$\,. This Stein result was known for a few years \cite{W3}, and in fact a somewhat more precise result is known \cite{WZ2} for the flag domains to which we apply the Barlet–Koziarz method, but the proof here is much more direct and holds the possibility of greater generality. Also, some of the tools developed here apply directly to open orbits that need not be measurable, avoiding separate arguments of reduction to the measurable case.

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Published 1 January 2000