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Remarks on homogeneous complex manifolds satisfying Levi conditions. (English) Zbl 1215.32015

After recalling the main ideas of Aldo Andreotti concerning his studies on homogeneous complex manifolds satisfying various types of Levi-conditions, the author studies the homogeneous complex manifolds which are called pseudoconvex or pseudoconcave. A connected complex manifold \(X\) is called pseudoconcave if \(X\) contains a relatively compact open set \(Z\) such that, for every point \(p\in\text{cl}(Z)\), there exists a holomorphic map \(\psi: \Delta\to\text{cl}(Z)\) of the unit disk \(\Delta\) in the complex plane \(\mathbb{C}\) with the property that \(\psi(0)= p\) and \(\psi(\text{bd}(\Delta))\subset Z\). A complex manifold \(X\) is called pseudoconvex if there exists an exhaustion function \(\rho: X\to\mathbb R\), which is plurisubharmonic outside a compact set in \(X\).
The author first clarifies the structure of complex Lie groups, and shows, among other things, that pseudoconcave connected complex Lie groups are compact complex tori. Next, he considers the nilmanifold \(X\), i.e., \(X= G/H\), where \(G\) is a connected complex nilpotent Lie group and \(H\) is a closed complex subgroup of \(G\). The author shows that pseudoconcave nilmanifolds are compact. Next, he defines the main objects of this article which are called flag domains. For that purpose, consider a real semisimple Lie group \(G_0\) and its complexification \(G\), and consider the case of flag manifolds \(Z= G/Q\) which is a compact \(G\)-homogeneous projective manifold; that is, the case of fundamental interest for studying the representation theory of \(G_0\). Now, by a result of J. A. Wolf [Bull. Am. Math. Soc. 75, 1121–1237 (1969; Zbl 0183.50901)], we know that \(G_0\) has only finitely many orbits on the flag manifold \(Z\), in particular, \(G_0\) has open orbits in \(Z\). Such an orbit is called by definition a flag domain.
The main purpose of this article is to give evidence for the following conjecture: Flag domains are either pseudoconvex or pseudoconcave. In fact, the author characterizes a flag domain to be pseudoconvex by several (equivalent) properties. For example, a flag domain \(D\) is characterized to be pseudoconvex by the property that \(D\) is holomorphically convex and that the Remmert reduction \(D/\!\!\sim\) is Hermitian symmetric, where the equivalence relation \(x\sim y\) for \(x,y\in D\) is defined by the condition that \(f(x)= f(y)\) for every holomorphic function \(f\) on \(D\).
In order to investigate the class of flag domains which are pseudoconcave, the author introduces the class of flag domains which are called connected by cycles, where cycles are defined as linear combinations of irreducible compact subvarieties in \(D\) with positive integer coefficients. First, he proves that cycle connected flag domains are not pseudoconvex, and then he tries to prove, in general, they are pseudoconcave. In fact, after introducing the notion of generically 1-connected flag domains, he succeeds to prove that generically 1-connected flag domains are pseudoconcave, which gives some evidence for the above conjecture to be true.

MSC:

32Q55 Topological aspects of complex manifolds
32F17 Other notions of convexity in relation to several complex variables
22E10 General properties and structure of complex Lie groups
32M10 Homogeneous complex manifolds

Citations:

Zbl 0183.50901
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