Levi’s Problem for Pseudoconvex Homogeneous Manifolds

Suppose G is a connected complex Lie group and H is a closed complex subgroup. Then there exists a closed complex subgroup J of G containing H such that the fibration π:G/H ⟶ G/J is the holomorphic reduction of G/H; i.e., G/J is holomorphically separable and O(G/H)≅ π*O(G/J). In this paper we prove that if G/H is pseudoconvex, i.e., if G/H admits a continuous plurisubharmonic exhaustion function, then G/J is Stein and J/H has no non-constant holomorphic functions.

Some properties of Grauert type surfaces

In a previous work, we classified weakly complete surfaces which admit a real analytic plurisubharmonic exhaustion function; we showed that, if they are not proper over a Stein space, then they admit a pluriharmonic function, with compact Levi-flat level sets foliated with dense complex leaves. We called these Grauert type surfaces. In this note, we investigate some properties of these surfaces. Namely, we prove that the only compact curves that can be contained in them are negative in the sense of Grauert and that the level sets of the pluriharmonic function are connected, thus completing the analogy with the Cartan–Remmert reduction of a holomorphically convex space. Moreover, in our classification theorem, we had to pass to a double cover to produce the pluriharmonic function; the last part of the present paper is devoted to the construction of an example where passing to a double cover cannot be avoided.

Positivity of metrics on conic neighborhoods of 1-convex submanifolds

Let [Formula: see text] be a holomorphic submersion from a complex manifold [Formula: see text] onto a 1-convex manifold [Formula: see text] with exceptional set [Formula: see text] and [Formula: see text] a holomorphic section. Let [Formula: see text] be a plurisubharmonic exhaustion function which is strictly plurisubharmonic on [Formula: see text] with [Formula: see text] For every holomorphic vector bundle [Formula: see text] there exists a neighborhood [Formula: see text] of [Formula: see text] for [Formula: see text] conic along [Formula: see text] such that [Formula: see text] can be endowed with Nakano strictly positive Hermitian metric. Let [Formula: see text] [Formula: see text] be a given holomorphic function. There exist finitely many bounded holomorphic vector fields defined on a Stein neighborhood [Formula: see text] of [Formula: see text] conic along [Formula: see text] with zeroes of arbitrary high order on [Formula: see text] and such that they generate [Formula: see text] Moreover, there exists a smaller neighborhood [Formula: see text] such that their flows remain in [Formula: see text] for sufficiently small times thus generating a local dominating spray.

Some Inequalities for the Omori-Yau Maximum Principle

We generalize A. Borbély’s condition for the conclusion of the Omori-Yau maximum principle for the Laplace operator on a complete Riemannian manifold to a second-order linear semielliptic operatorLwith bounded coefficients and no zeroth order term. Also, we consider a new sufficient condition for the existence of a tamed exhaustion function. From these results, we may remark that the existence of a tamed exhaustion function is more general than the hypotheses in the version of the Omori-Yau maximum principle that was given by A. Ratto, M. Rigoli, and A. G. Setti.

Parabolic Stein Manifolds

An open Riemann surface is called parabolic in case every bounded subharmonic function on it reduces to a constant. Several authors introduced seemingly different analogs of this notion for Stein manifolds of arbitrary dimension. In the first part of this note we compile these notions of parabolicity and give some immediate relations among these different definitions. In section 3 we relate some of these notions to the linear topological type of the Fréchet space of analytic functions on the given manifold. In section 4 we look at some examples and show, for example, that the complement of the zero set of a Weierstrass polynomial possesses a continuous plurisubharmonic exhaustion function that is maximal off a compact subset.

THE PLURICOMPLEX GREEN FUNCTION ON PSEUDOCONVEX DOMAINS WITH A SMOOTH BOUNDARY

In this article we deal with the behavior of the pluricomplex Green function GD(·;w), of a pseudoconvex domain D in [Formula: see text], when the pole tends to a boundary point. In [7], it was shown that, given a boundary point w0 of a hyperconvex domain D, then there is a pluripolar set E⊂D, such that lim sup w→w0 GD(z;w)=0 for z∈D\E. Under an additional assumption on D, that can be viewed as natural, one can avoid the pluripolar exceptional set. Our main result is that on a bounded domain [Formula: see text] that admits a Hoelder continuous plurisubharmonic exhaustion function ρ:D→[-1,0), the pluricomplex Green function GD(·,w) tends to zero uniformly on compact subsets of D, if the pole w tends to a boundary point w0 of D.

Some remarks on complex Lie groups

First we show that any complex Lie group is complete Kähler. Moreover we obtain a plurisubharmonic exhaustion function on a complex Lie group as follows. Let the real Lie algebra of a maximal compact real Lie subgroup K of a complex Lie group G. Put q := dimC Then we obtain that there exists a plurisubharmonic, strongly (q + 1)-pseudoconvex in the sense of Andreotti-Grauert and K-invariant exhaustion function on G.