Complex analytic properties of tubes over locally homogeneous hyperbolic affine manifolds (Q1070394)

Since the tangent bundle \(T_ M\) of any real affine manifold M admits a natural complex structure, it is of interest to compare properties of M and \(T_ M\). As an example of this philosophy, it is known that for domains M in \({\mathbb{R}}^ n\) that \(T_ M\) Stein is equivalent to M convex. In the present work affine manifolds M whose universal covering is affinely equivalent to a convex domain \(\Omega\) in \({\mathbb{R}}^ n\) containing no complete straight lines are considered. By means of known functions on a certain convex cone V(\(\Omega)\) associated to \(\Omega\) (e.g. the characteristic function of \textit{E. B. Vinberg} [Trans. Moscow Math. Soc. 1963, 340-403 (1965); translation from Trudy Moskov. mat. Obshch. 12, 303-358 (1963; Zbl 0138.433)] it is shown that one can construct a smooth strictly plurisubharmonic function \(\psi_ M\) on an open subset of \(T_ M\) whose complement is either the support of a divisor or else empty. If M is compact, then \(\psi_ M\) is an exhaustion function and if \(\Omega\) is homogeneous, then \(T_ M\) contains no positive-dimensional compact analytic subsets. With these assumptions it then follows that \(T_ M\) is Stein.