Pseudoconvex domains spread over complex homogeneous manifolds

Abstract

Using the concept of inner integral curves defined by Hirschowitz we generalize a recent result by Kim, Levenberg and Yamaguchi concerning the obstruction of a pseudoconvex domain spread over a complex homogeneous manifold to be Stein. This is then applied to study the holomorphic reduction of pseudoconvex complex homogeneous manifolds X=G/H. Under the assumption that G is solvable or reductive we prove that X is the total space of a G-equivariant holomorphic fiber bundle over a Stein manifold such that all holomorphic functions on the fiber are constant.Comment: 21 page

Levi’s Problem for Pseudoconvex Homogeneous Manifolds

“…Hirschowitz showed that a pseudoconvex, infinitesimally homogeneous X that does not contain any inner integral curve is Stein [12,Proposition 3.4]. This is the starting point of our investigations of pseudoconvex homogeneous manifolds that are not Stein given in [10]. By the maximum principle any plurisubharmonic function on a complex manifold X is constant along every inner integral curve in X.…”

“…One has to determine the "directions of degeneracy" of plurisubharmonic functions in terms of a certain subset of g whose corresponding holomorphic vector fields "kill them". So in [10] we define the Hirschowitz annihilator A to be the connected Lie subgroup of G whose Lie algebra is given by…”

“…For continuous functions the derivative is understood in the sense of distributions. By definition a is a complex vector subspace of g and it is also a Lie subalgebra of g that properly contains h, if X is not Stein [10,Lemma 3.3]. The corresponding (not necessarily closed) connected complex subgroup A of G contains the identity component H 0 of H and defines a complex foliation F = {F x } x∈X , the Levi foliation of the manifold X.…”

“…If the leaves of the foliation are closed, then they are compact and X is holomorphically convex [10,Theorem 4.1]. Indeed, the holomorphic reduction is then given by the Remmert reduction [26] and has compact fiber and Stein base.…”

Weakly complete compex surfaces

“…[7]). More recently, in [8], Gilligan, Miebach, and Oeljeklaus study the case of a pseudoconvex domain of any dimension, spread over a complex homogeneous manifold.…”